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MAA FOCUS December 2007

MAA FOCUS is published by the Mathematical Association of America in January, February, March, April, May/June, August/September, October, November, and December.

Editor: Fernando Gouvêa, Colby College; fqgouvea@colby.edu

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President: Joseph Gallian

First Vice President: Carl Pomerance, Second Vice President: Deanna Haunsperger, Secretary: Martha J. Siegel, Associate Secretary: James J. Tattersall, Treasurer: John W. Kenelly

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MAA FOCUS Editorial Board: Donald J. Albers; Robert Bradley; Joseph Gallian; Jacqueline Giles; Colm Mulcahy; Michael Orrison; Peter Renz; Sharon Cutler Ross; An-nie Selden; Hortensia Soto-Johnson; Peter Stanek; Ravi Vakil.

Letters to the editor should be addressed to Fernando Gouvêa, Colby College, Dept. of Mathematics, Waterville, ME 04901, or by email to fqgouvea@colby.edu.

Subscription and membership questions should be directed to the MAA Customer Service Center, 800-331-1622; email: maahq@maa.org; (301) 617-7800 (outside U.S. and Canada); fax: (301) 206-9789. MAA Headquarters: (202) 387-5200.

Copyright © 2007 by the Mathematical Association of America (Incorporated). Educational institutions may reproduce articles for their own use, but not for sale, provided that the following citation is used: “Reprinted with permission of MAA FOCUS, the newsmagazine of the Mathematical Association of America (Incorporated).”

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ISSN: 0731-2040; Printed in the United States of America.

MAA FOCUS

4 The Dangers of Dual Enrollment By David M. Bressoud

6 A Case Study of Concurrent Enrollment Partnership in Mathematics By Philip Cheifetz and Ellen Schmierer

9 Teaching Time Savers: The Exam Practically Wrote Itself! By Michael E. Orrison

10 The MAA Euler Study Tour Reports and Comments from the Participants

12 Eulogy of Leonhard Euler (1707-1783) By Siu Man Keung

14 AskingQuestions:ImprovingProspectiveTeachers’ Knowledge of Mathematics By Wendy A. Weber

16 First Annual TENSOR-SUMMA Grant Program a Success

17 TENSOR-MAA Grants for Women and Girls in Mathematics

19 Remember the Alamonoid: The Art of Factorization in Multiplicative Systems in San Antonio By Scott T. Chapman

21 Remembering Paul Cohen By Peter Sarnak

23 Statistics: A Key to Student Success in College and Life By John Loase

24 It is Time to Support Extremes By Reza Noubary

26 Employment Opportunities

Inside

Volume 27 Issue 9

On the cover: To escape the hurly-burly of the court in Berlin, Frederick the Great built himself a palace in Potsdam and named it “Sans Souci,” meaning “without a care.” There, Euler was asked to help create a fountain that would shoot water high up in the air. Unfortunately, the engineers did not heed his advice to test the ability of the tubes to withstand the required water pressure, and the fountain never worked successfully. (Photograph of Sans Souci Palace provided by Jim Smith)

MAA FOCUS

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December 2007

By Edward Aboufadel

The 2008 Annual Meeting of the American Association for the Advance-ment of Science will be February 14–18, in Boston, MA. This year’s program fea-tures many outstanding expository talks by prominent mathematicians, including a topical lecture by Curtis T. McMullen on “The Geometry of 3-Manifolds,” fo-cusing on the Poincaré Conjecture. The overall theme of this year’s meeting is “Science and Technology from a Global Perspective”; many of the symposia sponsored by Section A (Mathematics) are interdisciplinary sessions that fit this theme. The Section A symposia are only a few of the nearly 200 AAAS program offerings in the physical, life, social, and biological sciences. For further informa-tion, including the schedule of talks, go to http://www.aaasmeeting.org.

AAAS annual meetings are the show-cases of American science, and the participation by mathematicians and mathematics educators is encouraged. (AAAS acknowledges the generous contributions of AMS and MAA for travel support and SIAM for support of

media awareness.) The AAAS Program Committee is genuinely interested in offering symposia on pure and applied mathematical topics of current interest, and in previous years there have been symposia on subjects such as the chang-ing nature of mathematical proof, math-ematical ecology, and recent results in prime number research.

The 2009 meeting will be February 12-16, 2009 in Chicago. The Steering Committee for Section A seeks organiz-ers and speakers who can present sub-stantial new material in an accessible manner to a large scientific audience. All are invited to attend the Section A Committee business meeting in Boston on Friday, February 15, 2008, at 7:45 PM, where we will brainstorm ideas for symposia. In addition, I invite you to send me, and encourage your colleagues to send me, proposals for future AAAS annual meetings.

Edward Aboufadel is the Secretary of Section A of the AAAS. He can be reached at aboufade@gvsu.edu.

Mathematics at the AAAS, February 2008

Symposia Sponsored by Section A

Mathematics and the Brain (organized by Jack Cowan, Chicago)

Design of Mechanical Puzzles (organized by Peter Winkler, Dart-mouth)

Modeling the Dynamics of the Drug-Resistant Killers of the 21st Century (organized by Sally Blower, UCLA)

Quantum Information Theory (organized by Mary Beth Ruskai, Tufts)

New Techniques in the Evaluation and Prediction of Baseball Performance (organized by Edward Aboufadel, Grand Valley State)

Other Symposia of Interest to the Mathematical Community

Biometrics in Border Management: Grand Challenges for Security, Iden-tity, and Privacy

Atomic Detectives: Nuclear Forensics and Combating Illicit Trafficking

Ethical Issues in Scientific Publishing

Enhancing Science Globally Through High-Performance Computing and Simulation

Information, Computing, and Commu-nications: Keys to Sustainable Global Development

Collaboratively Developing Student Mathematical Thinking Among APEC Member Economies

Promoting the Success of Minority Graduate Students

Inside the Double Bind: Women of Color in Science, Technology, Engi-neering, and Mathematics

Steering Committee for Section A

Chair: Carl Pomerance (Dartmouth College)

Chair-Elect: William Jaco (Oklahoma State University)

Retiring Chair: Jack Cowan (University of Chicago)

Secretary: Edward Aboufadel (Grand Valley State University)

Members at Large:

Walter Craig (McMaster University)

Mary Beth Ruskai (Tufts University)

David Isaacson (Rensselaer Polytechnic Institute)

Claudia Neuhauser (University of Minnesota)

Looking for Hard Problems?

The world premiere of George Csicsery’s film Hard Problems: The Road to the World’s Toughest Math Contest will happen at the Joint Mathematics Meetings in January. See page 31 for more details.

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MAA FOCUS December 2007

The Dangers of Dual Enrollment

Dual or concurrent enrollment pro-grams are arrangements between a local college and high school whereby stu-dents enrolled in certain classes receive both high school and college credit. For the purposes of this article, the term will be used exclusively for classes that are taken at the high school and receive dual credit.

Though they have existed for a long time, dual enrollment programs have taken off in the past decade as high schools and colleges have come to see them as mu-tually beneficial. In fact, they seem to be a win for everyone: Colleges like them because, even at reduced tuition, they are a source of additional revenue and a way of increasing enrollment numbers at very low cost. High schools like them because of the prestige of offering col-lege-credit courses. Students like them because they eliminate the need to take a high stakes national test, such as the AP Calculus exam, in order to earn credit, and they look good on a transcript when applying to college. Parents like them because they can shorten the number of years their child will be in college, and thus the number of tuition payments.

I know that college material can be taught in high schools at a level that is compa-rable to what is taught in college. This is the premise of the College Board’s Advanced Placement Program, and it is justified in the high success rates of students who use the college credit they have earned to move directly on to more advanced courses. But not everyone who studies college-level material in high school achieves college-level pro-ficiency, as witnessed by the fact that at most a third of those who study calculus while in high school earn college credit for this course, and many of those who do not earn college-level credit struggle when they take a real college course on this same material. For this reason, a red flag goes up when colleges see students coming to them with credit for courses taken under dual enrollment programs.

What has been the level of accountabil-ity in these programs? How has the mas-tery of this material been assessed?

This column looks at data from the CBMS Survey, Fall 2005, references the example of Nassau Community College which has shown what it takes to build a good dual enrollment program (see also the separate article “A Case Study of Concurrent Enrollment Patnerships in Mathematics” in this issue), and de-scribes the Early College High School Initiative, a national dual enrollment program with both promise and danger.

CBMS Survey Data

While a small number of students take statistics under dual enrollment, most of the dual enrollment courses in mathe-matics are either college algebra/precal-culus or calculus. Just offering college algebra/precalculus as a college-credit course taught in a high school can be problematic. Many colleges, Macalester among them, no longer consider these to be college-level courses. Even colleges and universities that do offer college credit for these courses usually treat them as preliminary to the mathematics required for mathematically intensive majors. If these courses really are high school mathematics, what does it mean to offer college credit for them when they are taken in high school?

To get an idea of the annual numbers, the CBMS survey counted the number of students in dual enrollment programs in both spring and fall terms of 2005 [1]. There were 63,986 students studying college algebra/precalculus and 33,436 studying calculus. Both 2-year and 4-year colleges are involved in awarding dual enrollment credit: 22% of the stu-dents in college algebra/precalculus and 42% of those in calculus I received their credit from a 4-year college. Occasion-ally, these courses are taught by college faculty, but that is the exception. Only 4% of the courses affiliated with 4-year

colleges and 12% of those affiliated with 2-year colleges are taught by col-lege faculty. Even the label “college fac-ulty” can be problematic because there are cases where high school teachers are designated as faculty of the college.

College departments were asked to self-report on whether they maintained control over four aspects of these pro-grams: selection of textbook, syllabus, final exam, and choice of instructor. The table on page 5 shows the results. I find it very disturbing that the most effective means of ensuring quality control over the final examination and the choice of instructor, are so seldom used.

This lack of accountability can lead to serious problems for both the students and the colleges at which they are ac-cepted. While there are many excellent high school teachers capable of teaching these courses as well as or better than the instruction at college, and while even the term “college-level” is prob-lematic because of the great disparity among colleges in how such courses are taught, the goal of any dual enrollment course is that students who successfully complete it are at least as well prepared for the next course in the sequence as those who have taken it at the college from which credit is given. Without close oversight and supervision, there is danger that credit might be given even when the level of preparation is below that required for the next course.

Maintaining Quality

There are dual enrollment programs that do maintain the control that is needed. Nassau Community College on Long Island, New York, is one that is doing a good job. Further information on their program can be found elsewhere in this issue [2].

There now is an accreditation agency for dual enrollment programs, the Na-tional Alliance of Concurrent Enroll-

By David M. Bressoud

MAA FOCUS

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December 2007

ment Partnerships (NACEP) [3]. For those readers who know more about this organization, I would be very interested in your assessment of its usefulness. What impresses me most about NACEP is that its accreditation is accompanied by a rating system that describes the de-gree of control along several different dimensions. It appears that NACEP ac-creditation may be a useful tool to help colleges decide when dual enrollment credit should be accepted.

Early College High School (ECHS)

In learning about dual en-rollment pro-grams, I have also been alert-ed to a move-ment that has the potential to greatly impact the educational sys-tem as we know it. The Early College High School Initiative is a nation-wide collection of programs that award col-lege credit for courses taken while in high school. It is targeted specifically at “low-income youth, first-generation college goers, English language learn-ers, students of color, and other young people underrepresented in higher edu-cation.”[4] It is committed to instruction that is “rigorous, relevant, and relation-ship-based” (its 3R’s). Alarmingly, one of its primary goals is to allow these students to earn up to two full years of college credit while still in high school. Spearheaded by the Bill & Melinda Gates Foundation with support from Carnegie Corporation of New York, The Ford Foundation, and The W.K. Kellogg Foundation, it is a serious undertaking that is already operating in 130 schools, involving over 16,000 students. The goal by 2012 is 239 schools and 96,400 students.

Unlike the dual enrollment programs de-scribed above, most of the college-level classes (58% of all programs) are taught on college campuses, while another 22% are in special facilities devoted to ECHS. For the most part, the courses

that carry college credit are taught by college faculty (again, this can be an ambiguous designation), but usually to classes made up entirely of high school students. Of eight ECHS programs that were studied in detail in 2006, five teach at least some of the college-credit cours-es in classes that enroll both college and ECHS students, but only one of the pro-grams teaches all of its college-credit courses in this manner. [5, p. 33]

This makes a difference. In an indepen-dent study of this program undertaken during 2004–05, “interviews revealed that college instructors were more like-ly to maintain their usual standards if ECHS students were enrolled alongside other college students, rather than com-prising the entire class.” [5, p. 35] The program is new enough that as yet there are no reliable statistics on its effective-ness at getting at-risk students into and through college.

In an independent evaluation prepared for The Bill & Melinda Gates Founda-tion, the American Institute for Research noted that college faculty make a very conscious effort to maintain college-level expectations in these courses, but that “rigorous instruction was elusive [in the ECHS high school classes], par-ticularly in mathematics classes.” [5, p. 91] Of eight programs offering college credit that were studied in some detail in the spring of 2006, five did teach at least some classes that enrolled both college and high school students, but only one of the colleges did this exclusively.

As I see it, the primary goal of this ini-tiative is to introduce rigorous, college-level material to at-risk students, giving

them a sense of what college is like and what it will take to succeed while pro-viding an environment that helps them to succeed. This is a very worthy goal. I do worry about the promise of be-ing able to complete two years of col-lege-level courses while in high school. Students who successfully complete this program and arrive at college with the intention of graduating in only two more years will find themselves shut out of any mathematically intensive major.

Few students, even among the most talented and acceler-ated, arrive at college with the three semesters of calculus and the semester of linear alge-bra required before taking junior/senior

level courses in mathematics, physics, or chemistry.

The independent report does suggest backing off from the promise of two years of college credit earned while in high school as a “core principle” and replacing it with a commitment to com-press the time needed to complete a post-secondary degree. [5, p. 93] Even that commitment may make it more dif-ficult for these students to enter math-ematically intensive fields.

This initiative is already encountering some problems in maintaining rigor. I am concerned about what will happen to the expectations for achievement lev-els as this initiative expands and attracts imitators who sell their program as a way of cutting college costs.

Many forces have combined to drive both dual enrollment and the ECHS promise of a high school diploma that comes with an associate’s degree at-tached. High college tuition is one of them. Until now, demand for college degrees has seemed perfectly inelas-tic; people will pay whatever colleges charge. As everyone knows, that cannot be true. People are not going to stop go-

Level of control by college mathematics departments over dual-enrollment courses.

College Control Over

4-year colleges 2-year colleges

never sometimes always never sometimes alwaysTextbook 41% 15% 44% 14% 12% 74%Syllabus 2% 6% 92% 4% 7% 89%Final Exam 40% 30% 30% 36% 28% 37%Instructor 32% 20% 48% 35% 13% 52%

6

MAA FOCUS December 2007

ing to college, but they may well seek perverse and counter-productive routes to its completion. For those of us who worry about the low number of students going into mathematically intensive ma-jors, this should be a matter of concern.

References

[1] CBMS Survey, Fall 2005, tables SP.16 and SP.17, pages 62–63. http://www.ams.org/cbms/report05/

[2] Philip Cheifetz and Ellen Schmierer, A Case Study of Concurrent Enrollment Partnership in Mathematics, see below.

[3] National Alliance of Concurrent Enrollment Partnerships. http://www.nacep.org

[4] The Early College High School Ini-tiative. http://www.earlycolleges.org/

[5] American Institute for Research and

SRI International. Evaluation of the Early High School Initiative. Prepared for The Bill & Melinda Gates Founda-tion. 2007. http://www.earlycolleges.org/publications.html

David M. Bressoud teaches at Macales-ter College. He will be President-Elect of the MAA beginning in January 2008. A version of this article also appeared as a Launchings column in July, 2007 on MAA Online, at http://www.maa.org/columns/launchings.

By Philip Cheifetz and Ellen Schmierer

A concurrent enrollment partnership (CEP) is an association between a col-lege and one or more high schools that allow qualified high school students to earn college credit by taking college courses taught in their high schools. As reported by David Bressoud elsewhere in this issue [1], enrollment in such pro-grams has grown exponentially in the past few years.

This article reports on the Partnership Program in Mathematics at Nassau Community College (NCC) on Long Island, NY, as an example of what we believe the parameters of a success-ful CEP with integrity should embody. NCC, the largest single campus com-munity college in New York, serves Nassau County, whose population is over 1.3 million people. The College enrolls over 21,000 students per year. The Partnership Program in Mathemat-ics [2] began as an outreach program to local high schools in 1997 as a result of a bold redesigning of NCC’s calculus courses in 1993. At that time, the NCC mathematics department began teach-ing what was then referred to as “reform calculus.” In particular, the department adopted the calculus curriculum and materials developed under a National Science Foundation grant by the consor-tium based at Harvard University. Since NCC captured approximately 28% of all Nassau County college-bound seniors at that time, we felt in was incumbent upon

us to alert the high schools in the county of the dramatic changes their students would face if they enrolled in calculus courses at NCC.

In 1993, a workshop was offered to high school mathematics chairpersons and their department members. The calculus workshop consisted of two eight-hour days in the fall and spring semesters. These four workshops were attended by approximately 25 teachers, many of whom were department chairs. The at-tendees received hands-on instruction on the new “fewer but deeper” ideas in the NCC curriculum. A second set of four calculus workshops was given the following year to a new set of partici-pants.

With the success of the calculus reform, the Harvard Consortium developed a precalculus curriculum and associated materials. Again, NCC adopted this cur-riculum and ran four new workshops in precalculus. These workshops were at-tended by many of the same attendees who had attended the calculus work-shops.

During one of the sessions in 1996, one of the high school chairs asked if he could offer precalculus in his high school and have his students receive NCC cred-it. His reasoning was “you trained us, we will use the same textbook as NCC, we will use your exams, and we will

come to NCC once a month to discuss the progress of the partnership.” Meet-ings with the NCC administration and the College union were held at NCC to examine the viability and ramifications of such an arrangement. After a year of meetings, NCC began a pilot partner-ship program in precalculus in the fall of 1997 with two high schools and 47 students. The model we developed had the key components that follow. The last item was recently added.

• Any high school that wishes to par- ticipate in our CEP must send the teachers who would teach the course to a one-week summer workshop at NCC to prepare them to teach the course. Implicit in this preparation is the introduction to the NCC course curriculum and the textbook we use, as well as working out many of the homework exercises. • These teachers must agree to attend a regularly scheduled users’ group at NCC to discuss issues, examinations, upcoming material, etc. • School districts must agree that the high school class will be visited every second week by an NCC faculty member who will teach the next scheduled lesson to the high school class. • School districts must agree that the final course grade will be computed using examinations written and graded by faculty at NCC.

A Case Study of Concurrent Enrollment Partnership in Mathematics

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December 2007

• Students will be taught using The Way of Archimedes: formal definitions and procedures evolve from the investigation of practical problems. • The responsibility for learning will be gently shifted from teacher centered to student centered. • Faculty will continually encour- age students to adapt to ever changing challenges in their academic lives. Furthermore, the techniques for learning to meet these various challenges may have to change.• Course texts, syllabi and materials will reflect the spirit of the standards established by the National Council of Teachers of Mathematics (NCTM), the American Mathematical Association of Two-Year Colleges (AMATYC), and meet the recommendations of the Mathematical Association of America (MAA).• Course quality and standards should meet the recommendations published by The National Alliance of Concurrent Enrollment Partnerships [3] and endorsed by the Chancellor of the State University of New York (SUNY).

The first step in the implementation of our CEP is the one-week summer work-shop, which is open to all high school teachers in the county whether or not they wish to participate in our CEP. The workshop serves as a brush-up for many participants, as well as an exposure to the rigor and performance we expect.

After the completion of the course, each new participating high school teacher is introduced to his or her NCC partner professor. Telephone numbers and email addresses are exchanged and plans are made for the professor’s first visit. It should be noted that the NCC professor has this high school partnership course as part of his or her full-time schedule. At the first visit, the professor outlines the course expectations, hands out the NCC course outline, discusses policy, and informs the class that the NCC course-wide final examination (worth 35% of the final course grade) is given

at the College. Three other course ex-aminations (each worth 10% of the final grade) are given at the high school. All these exams are constructed and graded by the NCC professor. The remaining 35% of the final course grade is com-posed of the average of many quizzes and tests constructed by the high school teacher in collaboration with the NCC professor.

As we pioneered this CEP in 1997, the above four points seemed reasonable to all parties. However, today, ten years later, there is an overwhelming national pressure to pass all students and more likely than not, award every high school and college student a grade of A or B, which, in many CEPs, translates into the use of easier texts than on the college campus and a compromise on rigor and curriculum.

In [1], Bressoud has reported the data that shows how few colleges maintain control over textbook, syllabus, final exam, and instructor, Given this data, it appears that today the NCC program is rather uncommon, since we completely control textbook selection, course syl-labus, the final exam, and the NCC instructor. Our model does not depu-tize the high school teacher to assign the NCC final course grade, nor does the high school teacher have adjunct status in the NCC mathematics depart-ment. Rather, the high school teacher has the shared responsibility for impart-ing instruction and therein produces the partnership. Although the students who have enrolled in finite mathematics, pre-calculus or calculus are among the more mathematically talented students in their respective high schools, only 88% receive a C or higher.

An unexpected benefit of our regularly scheduled users’ group at NCC is the interaction between high school teach-ers and NCC faculty on topics that are tangentially related to the curriculum. The NCC professors learn about the awarding of significant partial credit, parental pressure, administrative pres-sure, and statewide mandates, while the high school teachers learn about the lack of student motivation, resistance to the shift from teacher centered to stu-

dent centered instruction, the multiple representations of concepts, the differ-ence between superficial and deep un-derstanding, and the shift from drill to interpretation and the absence of weeks of review for standardized exams. This interaction provides an appreciation for the issues of each group, and is very useful for establishing a collegial rela-tionship.

In 2006, 11 NCC instructors and 19 high school teachers were responsible for 504 enrollees in seven public high schools. This year, we expect over 600 students to enroll in one of our three mathematics courses. To date, over 3,500 high school students have been through the program. In 2000, the program was recognized by the American Mathematics Association of Two-Year Colleges and was awarded its Annenberg/CPB Foundation’s IN-PUT Award for “innovative curriculum and pedagogy in introductory college courses before calculus.”

References

[1] David Bressoud, The Dangers of Dual Enrollment. See page 4.

[2] The NCC Partnership Program in Mathematics. http://www.matcmp.sun-ynassau.edu/~cheifp/partners.htm

[3] National Alliance of Concurrent Enrollment Partnerships. http://www.nacep.org

Phil Cheifetz has been teaching at Nas-sau Community College (NCC) for 41 years. He is the co-author of five mathe-matics texts and numerous articles. Phil was a founder of AMATYC and served as its president in 1978. He is the co-director of the Partnership Program in Mathematics.

Ellen Schmierer has been teaching at NCC for 12 years. She is the co-author of three mathematics texts and has pre-sented many papers at both local and national mathematics meetings. Among her other duties, Ellen is co-director of the Partnership Program in Mathemat-ics and handles the day-to-day opera-tions of the program.

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MAA FOCUS December 2007

Found Math

Do you have to pick me up in your Acme uniform? It’s, like, soooooo embarrassing.”

“I certainly do,” I replied, giving her a big smoochy kiss to embarrass her further, something that didn’t really work, as the pupils in her math class were all grown up and too obsessed with number sets and parameterized elliptic curves to be bothered by a daughter’s embarrassment over her mother.

“They’re all a bit slow,” she said as we walked to the van. “Some of them can barely count.”

“Sweetheart, they are the finest minds in mathematics today; you should be happy that they’re coming to you for tutoring. It must have been a bit of a shock to the mathematics fraternity when you revealed that there were sixteen more odd numbers than even ones.”

“Seventeen,” she corrected me. “I thought of another one on the bus this morning. The odd-even disparity is the easy bit,” she explained. “The hard part is trying to explain that there actually is a highest number, a fact that tends to throw all work regarding infinite sets into a flat spin.”

Thursday Next in First Among Se-quels, by Jasper Fforde, p. 108.

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MAA FOCUS

9

December 2007

By Michael E. Orrison

When I first started teaching, creating an exam for my upper division courses was a genuinely exciting process. The material felt fresh and relatively unex-plored (at least by me), and I remember often feeling pleasantly overwhelmed with what seemed like a vast supply of intriguing and engrossing exam-ready problems. Crafting the perfect exam, one that was noticeably inviting, ex-ceedingly fair, and unavoidably illumi-nating, was a real joy.

As the years went by, however, the pro-cess of creating an exam began to feel more and more like a chore. What had once seemed like a vast supply of great problems now began to look like a nice, tidy, but ultimately small collection of simple and uninspiring questions. The excitement had worn off, and I was be-ginning to feel like I was spending too much time crafting or looking for prob-lems. I still wanted to create perfect ex-ams, though, so I decided that I needed to find a new source of inspiration.

So, a few years ago, in anticipation of what seemed like another uneventful exam making session, I decided to get my students involved in the process. A few days before I was to write the exam, I gave each student a sheet of paper that had, printed at the top, the following:

Problem Proposal: Propose a problem for the upcoming exam and explain why it should be on the exam.

I explained to them that (and I really do believe this) one characteristic of a good student is his/her ability to anticipate what an upcoming exam will look like. He/she should have a solid feel for the big ideas in a course, and be able to sug-gest several questions that could act as vehicles to demonstrate a thorough un-derstanding of those ideas. This exercise would, therefore, give them an opportu-nity to go through that process.

I made it clear that the exercise was com-pletely optional, and that there would be no “extra credit points” assigned to it. I did, however, tell them that I would carefully read each submission, and that there was a good chance that one or two of their problems would make their way onto the exam, especially if the justifi-cation was solid and compelling. I also told them that I thought this was a great way to begin studying for the exam, and that they were free to share their ideas with their classmates.

The response was wonderful. First of all, almost every student submitted a prob-lem together with a solid justification; from their responses, I could quickly get a sense of what they thought were the major themes of the course. Moreover, their justifications were thoughtful and, more often than not, made their point while cleverly connecting together sev-eral of those major themes.

From an exam writing perspective, it was impossible for me not to feel as though the exam was essentially writing itself as I read through their responses. The questions were at the right level, used the right notation, and were fo-cused on the right ideas. As you might expect, I found a lot of my old stand-bys in the mix, but they now seemed re-freshed and ready for action. It was like being handed a draft of the exam, and I was the editor. It was the inspirational spark I needed.

I now use the Problem Proposal sheet in all of my classes, and at least 90% of my students submit a problem each time. I have found that in addition to speeding up the process of creating an exam, it gives me an easy way to gauge student understanding at key points through-out the course. Moreover, the resulting exams seem to be ultimately more fair and connected to my classes because, in a very real sense, they are derived (or

at least inspired) by the classes them-selves. Thanks to my students, crafting the perfect exam is again a real joy.

Time spent: About one minute to read through each submission.

Time saved: About 30 to 90 minutes of creating or searching for exam ques-tions.

Michael Orrison teaches at Harvey Mudd College. He is the editor of the Teaching Time Savers series.

Teaching Time Savers:The Exam Practically Wrote Itself!

Teaching Time Savers are articles designed to share easy-to-implement activities for streamlining the day-to-day tasks of faculty members every-where. If you would like to share your favorite time savers with the readers of FOCUS, then send a separate email description of each activity to Michael Orrison at orrison@hmc.edu. Make sure to include a comment on “time spent” and “time saved” for each ac-tivity, and to include pictures and/or figures if at all possible.

The MAA makes it easy to change your address. Please inform the MAA Service Center about your change of address by using the electronic combined membership list at MAA Online http://www.maa.org) or call (800) 331-1622, fax (301) 206-9789, email: maaservice@maa.org, or mail to the MAA, PO Box 90973, Wash-ington, DC 20090.

Have You Moved?

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MAA FOCUS December 2007

Thoughts on the Euler TourSiu Man-Keung

For me the most worthwhile part of this Euler Study Tour was the op-portunity to read the many original manu-scripts from the pen of Euler now kept in the archives in St. Petersburg and in Berlin. The neat and beautiful handwriting, the meticulous and copious writing and the substantial content enhance my admiration and respect of this great mathematician and personality. While I walked on the spiral stone staircase and the floor of the place that the great man once worked in, the feeling was both strange and uplifting. I regarded it an honour to be able to read a short

eulogy before his tomb in the Alexander Nevsky Monastery Cemetery (see page 12).

Thanks to the kind arrangement of a Russian friend of Andrew Potter, a few of us went to visit the Pulkovo Observatory on a free afternoon. The Observatory is situated in a large bushy estate on the outskirts of St. Petersburg. Two things about the visit particularly impressed me. The first is the spirit of enthu-siasm and dedication the staff possess despite the not-so-afflu-ent condition (compared with some institutions which howev-er do not accomplish as much). When I saw the old-fashioned observatory towers hidden in the bush I thought at first they were only historical remnants, but after explanation by the staff I realized that those were still working well and had ac-complished good work in the past. In one telescope a part is actually a bicycle chain cleverly converted to fit the purpose. It reminds me of the kind of self-reliance practiced by Chinese scientists from the 1950s to the early 1980s to cope with the financially inadequate situation at the time, but in high spirits and with full enthusiasm.

The MAA Euler Study Tour

Man Keung Siu reading his eulogy to Euler.

The MAA’s Leonhard Euler Study Tour took place from July 1–14, 2007. The group of 26 intrepid travelers assembled initially in the An-dersen Hotel in Saint Peters-burg, Russia on July 1. Led by Victor and Phyllis Katz, the group then spent six days in the former Russian capital, visiting places where Euler lived and worked, as well as his tomb. We also were privileged to examine some of Euler’s notebooks and other original documents in the Archives of the Rus-sian Academy of Sciences. Naturally, we also visited some of the main tourist sites of St. Petersburg, and also learned about mathematics and mathematics education in Russia today.

On July 7, we flew to Basel, Euler’s birthplace. There, besides seeing the house where Euler grew up and the gymnasium that he attended, we also visited the houses and graves of several of the members of the Bernoulli family. In addition, we had a great time exploring the mathematics exhibit that had been originally put together for the ICM in Madrid in 2004.

On July 9, we went on to Berlin, the city where Euler spent close to a quarter of a century in the prime of his life in the employ of Fred-erick II (sometimes known, incorrectly according to one of our guides, as Frederick the Great). There we heard several lectures on Euler’s life and work while visiting the Academy of Sciences and the Humboldt Uni-versity, formerly the Uni-versity of Berlin. We were treated to another exhibit of Euler’s papers, including

the maps he worked on as part of his duties in the Academy. At the Technical University, we learned about German gradu-ate mathematical education today and had a tour of their vir-tual reality exhibit as well as their three-dimensional printing equipment. We concluded our tour with a visit to Frederick’s summer palace in Potsdam, where Euler had tried unsuccess-fully to persuade Frederick that he needed better materials in the pumping system if he wanted the fountains to work. Victor was responsible for putting together most of the academic por-tions of the tour, while Phyllis led several discussions on how to use the photographs and other information gleaned on the tour in teaching classes.

View of Basel

Reports and Comments from the Participants

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The second impressive thing is a copy of an old book that bears the scar of war in the defense of the motherland dur-ing WWII. The bullet went straight through the pages! At the time there was nothing but books to build a blockade for the Observatory under heavy German fire. I have heard and read about the 29-month siege of Leningrad (former name of St. Petersburg from 1924 to 1991), but this time I actually saw the relics of the siege for myself.

As someone interested in history of mathematics, the other (even older) books in the library certainly fascinated me also. Fortunately the staff succeeded in removing those precious old books to a safe place at the time of the siege; otherwise we would not be able to still read them. In particular, it was fascinating to read a copy of Astronomicum Caesareum (1540) written by Peter Apianus for Charles V, which is full of illus-trations in the form of movable hands-on devices, like those seen in a modern pop-up book for children!

The Euler AdventureCarole Lacampagne

The most exciting event of the Euler trip was our visit to the Euler Archives at the Russian Academy of Sciences. We were actually allowed to turn the pages of Euler’s notebooks and read his math-ematical ideas and formulas written in his own hand or that of his scribe. Other great experiences included hearing talks by Fred and Joel in the Old Aula in Museum lecture hall in Basil, surrounded by busts of Euler, the Bernoulli brothers, Gauss, and other famous mathematicians; and going to the Humboldt University in Berlin to hear lectures by Helmut Koch and Nor-bert Chapter on various aspects of Euler’s work.

Of course the sightseeing was also terrific. St. Petersburg is an absolutely beautiful city, filled with palaces and cathedrals, rivers and bridges. Basel retains its quaint charm with old buildings and the Rhine River running though it. And Berlin, with all its beautiful museums, still has a flavor of the devasta-tion of World War II.

And I did this marvelous trip on what I wore on the plane from Washington, a change of underwear, and a couple of borrowed clothes loaned to me by good friends! My luggage never ar-rived.

Two Weeks Studying the Life of EulerJohn Wilkins

Leonhard Euler was a re-markable mathematician. Although very few math-ematicians would disagree with this statement, that is not to say our community appreciates him fully. Par-ticipating in the tour was a chance to develop my appreciation of his work. One experience that left a significant impression on me was the day we visited the St. Pe-tersburg Academy. The room that contained Euler’s archived notebooks was small, so we had to take turns viewing them. One group toured the surrounding buildings, while the other viewed yet to be published manuscripts. To hold a notebook that Euler wrote over 250 years ago in his own hand was a very exciting moment. Some members of our talented group were able to translate a portion of the pages we were viewing from their original Latin and German. Those translations allow us the opportunity to discuss Euler’s thinking about a particular problem in mathematics. You can see photos of this experience at: http://www.maa.org/EulerTour/StP_Academy_Euler_Archives/index.htm.

This experience and many others have supplied me with a great wealth of material that I can use to enrich my History of Mathematics course.

A Non-Mathematical MomentMadeleine Long

The sky kept darkening as the rain interrupted the pace of the visit through the streets of Basel. As we stood in front of one of the many Euler sights, the group of twenty or so, uncomfortable yet in-tensely interested in the town and its many historical sights, our guide, clad in a bright red jacket that contrasted mark-edly with the gray black of the day, raised a question. Would it be ok, she asked, if she pointed out something that was totally non-mathematical, even non-scientific?

“Yes,” we all said, as if in a chorus. She then spoke of a phi-losopher she felt might be virtually unknown to most of us

Carole Lacampagne at the “Farewell to St. Petersburg” dinner with a bala-laika player.

Madeleine Long outside of the Hermitage in St. Petersburg.

John Wilkins at the Idiot Café in St. Petersburg.

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To commemorate the three- hundredth anniversary of the great Swiss mathematician Le-onhard Euler we would have to transport ourselves to the Eu-rope in the eighteenth century, which can be justly hailed as “the Age of Euler.” As the late Clifford Truesdell, himself a renowned mathematician and a life-long admirer of Euler, put it, “To study the work of Euler is to survey all the scientific life, and much of the intellectual life generally, of the central half of the eighteenth century.”1 It was a time when academic activities and scientific progress in Eu-rope took place most promi-nently in the few royal academies rather than in the many universities. It was also a time when no national distinction nor geographical boundary was recognized in the pursuit of science, so that eminent scholars came from different parts of Europe to study and work side by side

at the same academy. Among them was the “incomparable Leonhard Euler” (in the words of Johann Bernoulli).

Euler left his homeland at the age of twenty and spent the next fifty-six in-credibly productive years in the Russian

Eulogy of Leonhard Euler (1707–1783)By Siu Man Keung

Respectfully read before the tomb of Euler at the Alexander Nevsky Monastery Cemetery in St. Petersburg on July 3, 2007.

Academy in St. Petersburg and the Prussian Academy in Berlin. The French math-ematician-physicist François Arago said of him, “Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind.” The French philosopher-mathematician Marquis de Condorcet said in a eulogy of Euler, “He ceased to calculate and to breathe.” That happened on the evening of September 18, 1783 when Euler suffered from a stroke at his home in St. Petersburg. Today we come in homage to his burial

ground.

Euler is well known for his prolific output of an order of magnitude unpar-alleled in history, both in quality and quantity. There is this amusing story that the accumulation of the stack of his

but was evidently quite important to her, a man who had lived and worked in Basel longer than in any other place, a phi-losopher she was extremely eager, yet quite hesitant, to tell us about. Does anyone know who Nietzsche is? The responses that came from our group surprised and pleased our guide, and produced a radiance brighter than any we were to see that entire day. Mathematicians are interested in matters that pass beyond Euler and mathematicians! There is balance in the world after all!

ReflectionsontheEuler Tour, 2007Herb Kasube

The 2007 Euler Tour was inspiring as well as breathtaking. To look at Euler’s papers in the archive in St. Petersburg was a once in a lifetime opportunity. His notebooks were journals chronicling his life. What better way to look into the soul of an individual?

The charm of the city of Basel, Switzerland radiated from ev-

ery corner. The view from a 31st floor window was breathtaking. A ground level view of Basel re-vealed its old charm. I look forward to returning to Basel some day.

Being in Berlin conjured up memories of families separated by “The Wall” and thoughts about how far this city had come. One day, my morning run took me through the infamous “Check-point Charlie” in both directions. Forty years ago that would have been a lot more dangerous. Tears welled up in my eyes.

I end these reflections where they began, with the man of the year, Leonhard Euler. Standing next to his tomb In St. Peters-burg, I felt his presence.

Herb Kasube in front of Euler’s tomb.

Euler Notebook 6. Photograph by Herb Kasube.

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finished memoirs increased more rapidly than the rate these memoirs went to press, so that very often they were published in the reversed or-der of completion with the strange consequence that the content of a published mem-oir was more extended and improved than that of anoth-er that appeared later!

Approximately one-third of the research in mathematics, mathematical physics and engineering mechanics pub-lished in the last three-quar-ters of the eighteenth century was authored by Euler. The modern revision of his col-lected works (Opera Omnia) began in 1911 and is not yet finished, with al-ready near to eighty volumes published by 1994. In 1983 a special issue of Mathematics Magazine dedicated to his memory listed forty-four items of math-ematical terms or theorems that bear his name. His versatility as an all-round re-searcher is phenomenal, both in the con-tinuous and in the discrete, so that he was truly an expert in “con-crete” mathemat-ics!2 The influence of Euler goes far be-yond the eighteenth century and into the era we live in. Just to cite two examples. His idea on the famous seven bridges of Königsberg led to the Guan-Edmonds algorithm on the Chinese Postman Prob-lem in the early 1960s. It also led to the efficient Christofides algorithm in the mid-1970s that yields a solution to the (Euclidean) Travelling Salesman Prob-lem no worse than three-halves of the optimal solution. The second example is in number theory. Euler calculated the famous infinite sum of the reciprocal of consecutive squares, also known as ζ(2), to be π squared over six, and treated successfully ζ (n) for n even. More than 240 years later Roger Apéry proved that ζ (3) is irrational, but nobody yet knows what ζ (3) is. In recent years both in-terest and progress are rekindled in the investigation of (n) for odd n.

As an author and teacher Euler is noted for his clarity in exposition, but more than that he was eager to share with his readers his ideas that led to those

discoveries.3 No wonder Pierre-Simon Laplace said of him, “Read Euler, read Euler. He is the master of us all.” His primary interest was in the wonder of discovery and its explication, but cared little whether he or somebody else made the discovery. The late Russain historian of mathematics, Adolf P. Youschkevitch, described Euler as a well disposed man not given to envy, and further borrowed from the famous eulogy by Bernard le Bouyer de Fontenelle on Gottfried Wil-helm Leibniz to highlight this noble trait of Euler, “He was glad to observe the flowering in other gardens of plants whose seeds he provided.”4

One more thing about the man that earns our admiration is the way he led a fruit-ful life despite adversity. He was seri-ously ill at twenty-eight that led to loss of sight in his right eye at thirty-one. He was struck by illness again at fifty-nine that led to partial loss of sight in his left eye, and at sixty-four became totally blind for the last twelve years of his life. At sixty-four he was further hit by mis-fortune in losing his house and proper-ties in a fire, followed two years later by the death of his first wife. Euler had thir-teen children, but eight died in infancy. He took all the misfortune in stride and was so dedicated to his work that nearly half of his research was produced when he was close to sixty.

In the ancient Chinese Book of Odes one reads the verses that have come to con-note the noble character we admire and

1. C. Truesdell, Leonhard Euler, Su-preme Geometer, in C. Truesdell, An Idi-ot’s Fugitive Essays on Science, Spring-er-Verlag, New York, 1984, 337-379.

2. Concrete Mathematics is the title of the book written by R.L. Graham, D.E. Knuth, O. Patashnik, published by Ad-dison-Wesley in 1989. The authors say that they are not bold enough to try “dis-tinuous mathematics!”

3. Many more examples, besides the computation of ζ (2), can be found in M.K. Siu, Euler and heuristic reasoning; Mathematical thinking and history of mathematics, in Learn From the Mas-ters! Proceedings of the Kristiansand Conference on History of Mathematics and its Place in Teaching, August 1988, edited by F. Swetz et al, Mathematical Association of America, Washington D.C., 1995, 256-275; 279-282.

4. A.P. Youschkevitch, Euler, in Bio-graphical Dictionary of Mathemati-cians, Volume 2 (from Dictionary of Scientific Biography, Volume 4), edited by C.C. Gillispie, Scriber’s, New York, 1970-80, 736-753.

the virtuous deed we emu-late : “The high mountain I look up at it. The great road I travel on it.” Euler was a genius, whose height very few can hope to reach. But we can all learn from his great zest for life, work and study, his insatiable curiosity to know and to probe, his determination to procure deeper and deeper understanding, his indus-try, his modesty, his gener-osity, and his toughness in facing adversity with tran-quility. These are qualities we strive for.

The inscription on a plaque put up in his memory in his hometown (Riehen near Basel) sums up succinctly the life of this simple yet great man, “Er war ein grosser Gelehrter und ein güt-iger Mensch (He was a great scholar and a kind man).”

Instruments. Photograph by Phil Scalisi.

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Asking Questions:Improving Prospective Teachers’ Knowledge of MathematicsBy Wendy A. Weber

In 1987 Richard J. Crouse and Clif-ford W. Sloyer published the (now out of print) two-volume book Mathemati-cal Questions from the Classroom (Jan-son Publications, Inc.). Using this book as a guide, I designed a one semester hour course focused on the mathemat-ics needed for teaching middle and high school. This course helps prospective teachers deepen their own understand-ing of fundamental mathematics, make connections between the courses they have taken as mathematics majors, and learn how to handle questions that arise in the classroom. We explore questions from the middle school level, introduc-tory through advanced algebra, geom-etry, trigonometry, functions, probabil-ity, and calculus (statistics and discrete mathematics are undeveloped areas in the original text).

Course Design

The questions we explore could certain-ly fill more than one semester hour. We nevertheless chose to make the course only one semester hour, in part because our students already take a full load of courses from our department as well as the education department. We did not want to overburden them with more hours. In addition, because all math-ematics faculty already have a full load and we did not want to reduce offerings that all majors could take, we restricted the number of hours faculty would be teaching.

The structure of the course is simple. We meet once a week for fifty minutes. We either start class with a short quiz containing questions similar to ones we discussed in the previous class, or we begin with student presentations of the questions. In these presentations, stu-dents are expected to respond as if they were actually asked such a question by a student in one of their classes. Students know ahead of time which questions they will need to prepare, and I encour-

age them to discuss their answers with me before their presentation.

Most of the class time is spent discuss-ing the questions. Students are asked to read through the questions before com-ing to class and to think about how to answer them. I intentionally choose questions that are rich and offer extend-ed conversation. We typically spend only 2–3 minutes per question, though some questions (like those included be-low) are so rich they take up to 10 min-utes to cover. As students become more proficient at unpacking their knowledge, many questions that would have taken several minutes at the beginning of the course only take 30 seconds to discuss. In a 50 minute class period we can usu-ally cover 20–25 questions.

After each class students are expected to write solutions to all the questions discussed from the day. I periodically collect the solutions and examine them for accuracy. I make comments and cor-rections on a separate piece of paper so that students may fix their errors before they turn in all solutions at the end of the semester.

The Questions

We begin with questions from the mid-dle school level; understanding these questions provides a foundation for sub-sequent questions. Mathematics majors know and can perform fundamental al-gorithms, but they no longer question why they do what they do. These ques-tions help them unpack their knowledge and rebuild their understanding into a usable form for teaching mathematics.

Examples of questions we explore fol-low. To illustrate the depth of questions and the type of conversation these ques-tions can generate, I have included in brackets after the original question the warm-up or follow-up questions I might ask.

A student asks you the following ques-tion, “When multiplying decimals, why do we count off decimal places from right to left in each factor and then add them in order to find out where to put the decimal point in the answer?” How do you respond?

[What is a factor? What are other repre-sentations of decimals? When might one representation of a decimal (or a frac-tion) be more useful than another? Why do we line up decimal points when add-ing and subtracting decimals? Why do we move decimal places when perform-ing long division of decimals?]

In response to your question of how to define yk, a student says that y is mul-tiplied by itself k times. How do you re-ply?

[What does the student mean versus what s/he says? How could you deter-mine if the student has the correct idea and is only explaining it incorrectly? Does the context of the question make a difference as to whether you pick up this thread with the student and correct him/her?]

Tony can’t understand why division by zero is undefined. He wants 0 ÷ 0 = 0. His explanation is: “If you have nothing in one hand and you divide it by noth-ing, why don’t you still end up with noth-ing?”

[What is the operation of division? Is “zero” the same as “nothing?” What is 0 ÷ 0? How is this problem (or your ex-planation) the same or different from 1 ÷ 0?]

“If it is true that the square root of 25 is ± 5, then why doesn’t √25 = ± 5?”

[What does the square root symbol mean? What is a function? How do we solve x2 = 25? What are we really do-ing when we take the square root of both sides of this equation? What is √(x2)?]

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The students in your class are asked to solve the following problem. “Find the roots of x3 = 2x2 + 8x.” One student does the following work: x3 = 2x2 + 8x → x2

– 2x – 8 = 0 → (x – 4)(x + 2) = 0 → x = 4 or x = -2. Is the student correct?

[How many solutions should the equa-tion have? Why can’t s/he divide by x?]

A student hands in the following work: (6t)√3 = (6√3)(t√3). How would you help this student?

[What property is s/he attempting to use? How would you help this student see that his/her work is incorrect? How would you help this student see that multiplication distributes over addition and subtraction but not over multiplica-tion?]

How do you know for sure that some ir-rational number does not have a repeat-ing decimal eventually?

[What is an irrational number? A ratio-nal number? What is a series that con-verges to a rational number? How can we write any repeating decimal as a rational number? Why are we allowed to do these operations? What happens if

you assume an irrational number even-tually repeats?]

What is the difference between the terms polynomial, polynomial equation and polynomial function?

[How important is the language we use as teachers?]

Does lim x → 0 (1/x) = ∞ or does this limit not exist?

[What does the symbol ∞ mean? Is there a difference between saying a limit equals infinity and it does not exist? Is there always a difference?]

Why is an inverse function a reflection about the line y = x?

[Geometrically, what is a reflection? What is an inverse function? If the origi-nal point is (x,y) and the reflected point is (y,x), how do we find the line of re-flection and what is it?]

You ask your students to find the prob-ability of both children being boys in a two child family. One student says the probability is 1/3; another says it is 1/4. Is either student correct?

[What does a probability of 1/3 (or 1/4) mean in this situation? How can you il-lustrate the solution?]

A student asks you why sin2(x) + cos2(x) = 1.

[What is the Cartesian equation of the unit circle? What are the functions sine and cosine? This question is not found in Crouse and Sloyer’s text, but we be-lieve it is needed.]

The class has been very successful. It helps prospective teachers learn to draw on their considerable mathematical knowledge to accurately answer student questions at the correct level. Students enter their student teaching experience armed with 200 questions that cover nearly all the topics they will teach. More importantly, they enter the teach-ing profession able to answer their stu-dents’ questions with much more than a simple “because.”

Wendy A. Weber is Associate Professor of Mathematics at Central College in Pella, Iowa. She can be reached at we-berw@central.edu.

Calculators at the Smithsonian

On September 25, 2007, in commemoration of the 40th anniversary of its invention of the handheld calculator, Texas Instru-ments donated one of the original handhelds to the Smithsonian. Speakers at the donation ceremony included Dr. Brent D. Glass, Director of the National Museum of American History, Melendy Lovett, President of Texas Instruments Education Technology Division, and Jerry Merryman, one of the three inventors. The original, along with an array of the calculators that followed, was on display. Photographs and caption courtesy of Robert Vallin.

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The first TENSOR-SUMMA grants were awarded in April 2007, thanks to the generous support of the Tensor Foundation. The awards program was designed to provide grants to support college or university faculty conduct-ing programs to encourage the pursuit and enjoyment of mathematics among middle, high school, and beginning col-lege students from groups traditionally under-represented in mathematics. The quality of the proposals submitted was very high and had strong relevance to the goal of the grant program: engaging more minority students in activities that potentially open doors into mathemati-cally based professions. Twelve propos-als were recommended for funding.

Two awards were given for Math Cir-cles to serve inner city school students: the DC and the Oakland Math Circles. Project Director Jamylle Carter, of San Francisco State University, reported that the first five weeks of the Oakland Math Circle were spent on plane symmetries and tessellations. A seventh grade boy from the Circle said, “I love everything you’re doing because it’s stuff I’ve never seen before. It rocked!” Project Direc-tor Dan Ullman of George Washington University is conducting his Math Cir-cle at the Carriage House of the MAA. DC middle school students have a lively time experimenting, conjecturing, find-ing counterexamples, debating, proving, and generalizing. A grant for another Math Circle went to Joseph Meyinsse from Southern University, an HBCU.

Projects from the University of Texas at San Antonio, the University of Ari-

zona, and North Carolina A&T State University (an HBCU) are concerned with the transition from high school to college. The University of Texas at San Antonio’s Number Olympics was held in September. Project director Eduardo Duenez reports that they attracted a diverse pool of almost 50 participants. Duenez recruited UTSA freshmen at orientations days in his “Tensor Hut” with currency in TENSOR bills which came in denominations of 101 and 1010 tensors. Students earned tensors by solv-ing problems and used them to purchase prizes.

The University of Arizona’s David Sav-itt and William Yslas Velez held a one-week, four hour-a-day workshop for entering calculus students. Twenty-one students participated. An added bonus was that many of the participants en-joyed the interaction with the math ma-jors so much that they chose to major in mathematics! North Carolina A&T State University’s project director Alexandra Kurepa is also looking to the transition from high school and college. Kurepa has designed a program to provide en-richment activities for high school stu-dents enrolled in AP or Honors Calculus and precalculus as well as for students in the Early College program. She says that students from the Early College program find the experience especially helpful with their transition from high school to college.

Wentworth Institute of Technology’s Amanda Hattaway directs Students Loving Adventures in Mathematics (SLAM), a yearlong, twice a month Sat-

urday morning program designed to ex-pose high school sophomores to math-ematics in fun and innovative ways. As part of the SLAM Math Career Series, three actuaries from the International Society of Black Actuaries, a Boston university bioinformatics professor, and an information technology special-ist spoke to the students who will later present a poster session on these topics.

Bemidji State University’s Colleen Liv-ingston and Fort Lewis College’s Carl Lienert are both holding week-long summer mathematics camps for middle or high school students, many of whom come from near-by Indian reservations.

Fayetteville State University, an HBCU, the University of Minnesota, and Mount Holyoke College received grants to sup-plement existing successful programs. Fayetteville’s Project Director Kim-berly Smith Burton is adding a techni-cal literacy component to her Saturday Academy for middle school students. In September their teachers received in-struction in using the National Science Digital Library which they will pass on to their students. The University of Minnesota’s Project Director Harvey Keynes and Mount Holyoke College’s Project Director D. James Morrow are making a concerted effort to attract more young women of color to their summer program programs.

The deadline for receipt of the second annual TENSOR-SUMMA Grant ap-plications is February 15, 2008. All in-novative projects are welcomed. See the MAA website for details.

First Annual TENSOR–SUMMA Grant Program a SuccessBy Efraim Armendariz and Carole Lacampagne

Found Ego-Boost

Today I discovered the Public Library of Science journals, free, open-access academic journals in Biology, Medicine, Computational Biology (Sweet!), Pathogens, Neglected Tropical Diseases, and Clinical Trials. For once Google was not my informer, but FOCUS (the MAA newsmagazine).

—Eric “SigmaX” ScottThinking Aloud, 11/07/2007.

http://sigmax.no-ip.org/blog/?p=966

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In April 2007, the TENSOR-MAA pro-gram made grants of up to $6,000 to 17 institutions for faculty members to con-duct programs to encourage women and girls to continue the study of mathemat-ics. Among these, seven were renewals of previously funded programs.

Renewal Grants

The Joy of Thinking girls’ math club pro-gram at Texas Tech University is now in its fifth year and can boast a record num-ber of seven schools participating in fall 2007. Five of these schools have student populations almost entirely members of underrepresented groups. Project Direc-tor, Jerry Dwyer, wrote of his work in FOCUS, January 2006.

The MPower program at Metropolitan State University (Saint Paul, MN) of-fered its second annual, five-day summer camp to 21 middle school girls living in Saint Paul. Director Rikki Wagstrom is preparing to reunite these girls for a day of mathematics in November 2007.

The TENSOR-MAA program usually makes one grant for a Math Day. In 2006 and again in 2007, Hood College hosted their Sonya Kovalevsky Day for local high school girls. Betty Mayfield will direct the next day March 10th, 2008. Members of the AWM Student Chapter help with the day of workshops, panels and fun.

This is the second year of the Piney Woods Lecture Series in Mathematics, a program which invites well-known female mathematicians to campus to present talks which are accessible to undergraduates. The women then meet informally with students during a recep-tion, and have dinner with nominated female students. In the past 18 months there were five speakers and four are scheduled for spring 2008. More infor-mation is available at http://www.shsu.edu/~mth_jaj/pwls/ or by contacting project director, Jacqueline Jensen.

Another summer program is in its sixth year: The University of St. Thomas in St.

Paul, Minnesota, held its annual summer residential camp for sixteen high school girls June 24-29, 2007. Activities under the direction of Lisa Rezac, included math mini-courses, talks on connections in biology and mathematics, symmetry, topology and mathematical sculpture as well as a career panel and a trip to the Science Museum of Minnesota to see the calculus exhibit.

CSU Stanislaus, under the direction of Viji Sundar, conducted a week-long in-tensive Summer Academy called Prepar-ing Women for Mathematical Modeling- Robotics (PWMMR) which included a one day field trip to The Tech Museum of Innovation in San Jose. There will be a one-day follow-up in March 2008 when the participants will return to cam-pus to participate in a math conference on “Mathematical Moments” based on the AMS series.

The Girls Math & Technology Camp was held July 22-28, 2007, on the Uni-versity of Nevada, Reno campus for 60 middle school girls of diverse back-grounds from across Northern Nevada. A web site will provide participants and their parents year-round support and encouragement in math and technology from director, Lynda Wiest.

New Grants

The ten grants for new programs were made. These programs range in scope as their directors seek new ways to reach candidates who would have benefit from this work.

Rachelle De Coste (now at Wheaton College), recognized a need for assis-tance in the transition year from com-pleting the dissertation to gaining em-ployment. With the help of the Military Academy where she was employed she conducted a highly successful program in August 2007 which brought together at West Point ten women participants, five junior faculty “mentors” in addi-tion to senior faculty mentors from West Point for a two day conference. They will reconvene at the Joint Meetings

with plans for another CaMeW (Career Mentoring Workshop) for women fin-ishing their PhD’s in mathematics.

An entirely different program was held for a group of 50 high school women at Bloomsburg University of Pennsylva-nia, each of whom was given a scholar-ship for the First Annual Computer Fo-rensics Summer Experience for Young Women (CFSEYW). In this week-long program directed by Elizabeth Mauch, young women were introduced to the mathematics and computer science necessary for the field of computer fo-rensics by studying numerical analysis, statistics, mathematical modeling, and finite mathematics.

For many years Hampshire College has had a highly successful summer pro-gram for pre-college students who love mathematics. In an effort to enroll more women, HCSSiM recruited its female alumnae to mail customized postcards to recruit new girls to apply to HCSSiM 2007. Under the direction of sarah-ma-rie belcastro, they held mathematical and discussion activities with alumnae during the program to enhance the pro-gram for girls.

The first day of the USD Math Days for Women program was in October 2007 at the University of South Dakota, Ver-million SD; a second day will be in De-cember and the third in February. The young women participants from the sur-rounding area high schools worked on origami constructions and Geometer’s Sketchpad exercises in addition to learn-ing about the usefulness of mathematics in various careers. Communication be-tween the director, Violetta Vasilevska, program leaders, students and their teachers is by email where weekly prob-lems are posed.

At the University of California Santa Barbara, graduate students are making plans to attend the Joint Meetings in San Diego utilizing some of the grant funds. The group, let by Maria Isabel Bueno Cachadina and Birge Huisgen-Zimmer-mann, organized activities in the fall of

TENSOR-MAA Grants for Women and Girls in Mathematics

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2007 which included a documentary film showing of The Gender Chip, five professors talking about their research and, of course, accompanying social ac-tivities one of which was with women undergraduates.

Catherine O’Neil directed a new pro-gram at Barnard College which funded a day of mathematics and physics in partnership with the nearby Urban As-sembly Institute for Math & Science for Young Women. The entire school con-sisting of 78 sixth grade girls came to Barnard campus for a day and listened to lectures, did puzzles, won prizes, and competed in a group math competition for tickets to Six Flags.

Fifteen Hispanic and economically dis-advantaged young girls, Grades five and six, were granted scholarships to partici-pate in a two-week long summer Math Camp at Texas State University San Marcos. The girls were challenged to think and do mathematics on their own, and demonstrated a significant increase

in their algebra preparedness under the guidance of project director, Alejandra Sorto.

At the time this report was written the second three-day residential workshop planned by Janet Kaahwa to refresh 20 teachers of mathematics from ten schools in three districts (Masindi, Kibaale, and Hoima) in Uganda was completed. The facilitators were six women mathema-ticians who acted as role models af-ter participating in the first workshop held at Makerere University, School of Education, Department of Science, and Technical Education.

The Infinite Possibilities Conference (IPC) sponsored by the non-profit Build-ing Diversity in Science, North Carolina State University, and the Statistical and Applied Mathematical Sciences Insti-tute (SAMSI) directed by Kimberly Weems was held at the North Carolina State University, November 2–3, 2007. This conference was designed to foster increased participation of underrepre-

sented minority women in the math-ematical sciences. Conference activi-ties included plenary talks by Iris Mack and Freda Porter, research talks, student poster presentations, roundtable discus-sions, and panels on graduate studies and professional development along with workshops for high school stu-dents and the development of mentoring circles at every stage of the educational and professional pipeline.

In Fall 2007, under the leadership of Jacqueline Dewar, Women and Math-ematics for Future Teachers collabo-rated with various programs at Loyola Marymount University (the future K-8 teacher preparation program, Women’s Studies, Honors program) to cross-list a team-taught course on women and mathematics in Spring 2008. This course will examine historical and cur-rent equity issues in mathematics educa-tion and math-related careers through a study of the biographies and mathemati-cal contributions of nine women mathe-maticians from the 4th through the 20th centuries.

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December 2007

By Scott T. Chapman

In a typical beginning abstract algebra class, students are in-troduced to non-unique factorizations into products of irreduc-ibles by the example

6 2 3 1 5 1 5= ⋅ = +( )( )– – –

in the ring D =

Z –5 . Here the elements 2, 3, 1,+

– , – –5 1 5 are all irreducible and are pairwise nonassociate, i.e., the two different factorizations are not generated simply by multiplying factors by invertible elements. A careful analy-sis of this example shows that 1 5 1 5+( )( )– – – and 2 3⋅ are, up to associates, the only factorizations of into products of irreducibles in D. Moreover, D has the following nice proper-ty: if x is a nonzero nonunit of D and a a a b b bn m1 2 1 2, , , , , , ,K K

irreducible elements of D with x a a b bn m= =1 1L L , then n = m .

It is observations such as these which led mathematicians to become interested in the behavior of non-unique factorizations in other mathematical structures. Problems which involve non-unique factorizations of elements in integral domains and monoids have become frequent topics in undergraduate and graduate mathematics over the past 35 years. Solving such problems requires techniques from multiple areas of algebra and discrete mathematics.

A group of 33 participants gathered for five days from May 20th to May 25th 2007 at Trinity University in San Antonio, Texas for an MAA-PREP Workshop entitled The Art of Fac-torization in Multiplicative Structures. The workshop, unlike a regular research conference, had multiple goals:

• To present in accessible fashion the fundamental results of the theory of non-unique factorizations.

• To demonstrate that such results involve not only commu-tative algebra, but also rational and additive number theory, combinatorics, discrete structures and the geometry of Nk .

• To provide to college and university level faculty and gradu-ate students a rich source of material which not only can sup-plement existing courses in algebra, number theory and com-binatorics, but also can be taught as a stand-alone advanced undergraduate course or as an introductory graduate level course.

• To demonstrate that while advanced results in this area in-volve deep mathematics, many problems are suitable for mas-

ters level work, summer REUs, or even capstone undergradu-ate projects.

In keeping with the theme of the workshop, the participants represented a broad mathematical spectrum: about 60% col-lege or university faculty and 40% undergraduate or graduate students. Trinity University’s Holt Conference Center offered an intimate setting for the meeting.

Why might this topic be worthy of such a general gathering of interested mathematicians? Since 1988 several hundred papers dedicated to non-unique factorizations in commutative rings or monoid have appeared in refereed mathematical journals or conference proceedings (see the references in [4] for a complete listing). Two major conference proceedings ([1] and [3] ) dedicated primarily to factorization problems have been recently published by Marcel Dekker and Chapman & Hall. Additionally, 2006 has seen the publication of a major mono-graph by Geroldinger and Halter-Koch [4] which details the current state of research in this field.

The workshop began with keynote addresses by the workshop leaders, Scott Chapman of Trinity University and Jim Coyk-endall of North Dakota State University. The morning sessions consisted of a series of lectures by three leading researchers in this area: David Anderson from the University of Tennessee at Knoxville, Alfred Geroldinger from Karl-Franzens-Univer-sität in Graz, Austria, and Ulrich Krause from Universität Bre-men in Bremen, Germany. In combination, the main speakers offered over 80 years of classroom experience, in addition to over 200 refereed publications. The afternoon sessions includ-ed seminars by two graduate assistants, Paul Baginski from the University of California at Berkeley and Amanda Matson from North Dakota State University, who provided extensive experience in working with factorization problems at both the undergraduate and graduate level. The afternoons concluded with a series of problem/discussion sessions which not only addressed problems posed by the main speakers, but also how factorization material can best be used in the classroom and in undergraduate research projects.

The three main speakers focused on four different approaches to factorization theory prevalent in the literature today.

Algebraic: How do ring properties influence factorizations? Do polynomial rings, power series rings and other such struc-tures inherit the factorization properties of their base rings?

Remember the Alamonoid: The Art of Factorization in Multiplicative Systems in San Antonio

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Combinatorial: An in-depth analysis of the factorization properties of arithmetically motivated monoids (such as sub-monoids of the natural numbers). A study of the connection of factorization problems to problems involving zero-sum theory in finite abelian groups. This includes a study of the calcula-tion of the elusive Davenport Constant and cross number.

Geometric A study of factorization properties in submonoids of Nk . An exploration of the connection between factorization properties of integral domains and the structure of cones in Nk.

Number Theoretic: How do number theoretic properties in-fluence factorizations? What role do factorization properties play in some classical problems from number theory (e.g. rep-resentation of integers by quadratic forms)? How completely are orders in rings of integers determined by only considering factorization properties?

We offer the reader one example of the combinatorial/number theoretic aspects described above. This approach to factoriza-tion theory studies combinatorial sets and constants associ-ated to particular multiplicative structures. Suppose M is an monoid in which every element has an irreducible factoriza-tion. If x M∈ is not a unit, then the set of lengths of x is the set of integers n such that x can be factored as a product of n irre-ducibles. Write these lengths in increasing order: {n1,n2,…,nk}. We can look at several ways of describing this set. We define l x n( ) = 1 to be the length of the shortest factorization and let L(x) = nk to be the longest. We define ∆ x( ) to be the set of differences ni – ni–1, and ∆ M( ) to be the union of all the sets of differences as x ranges through the non-units in M.

The elasticity of an element x M∈ , denoted ρ x( ) , is the quotient L x x( ) ( )/ l . The elasticity of M is then defined as

( ) ( ) xsup{ \ }M x x M Mρ ρ= ∈ We say that M has accepted elasticity if there exists x M∈ such that ρ x( ) = ρ M( ) .

Determining these sets and constants, even for simple monoids, can be tricky, as we demonstrate with the set of integers S = {0,2,3,4,5,6,7,…} under regular addition. Notice that 2 and 3 are the only irreducible elements of S. Hence, an irreducible factorization of n S∈ is of the form n x n= ⋅ + ⋅

1 22 3. Factor-

izations in S are far from unique; for example,

17 7 2 1 3 4 2 3 3 1 2 5 3= ⋅ + ⋅ = ⋅ + ⋅ = ⋅ + ⋅ .

Thus,

l 17 6( ) = , L(17) =8. ∆ 17 1( ) = { } and ρ 17 8 6 4 3( ) = = .

Notice that the longest factorization of an element n S∈ con-tains the most possible copies of 2 and the shortest the most possible copies of 3. Hence, modulo 6 the exact values of l m( )

and L(m) can be computed with elementary number theory as seen in the table below. It turns out (and isn’t hard to check) that ρ m( ) ≤ 3/2 for all m and ρ m( ) = 3 2 if m ≡ ( )0 6mod . Thus ρ S( ) = 3 2 and the elasticity is accepted.

The workshop’s website, located at http://www.trinity.edu/schapman/ArtofFactorization.htm, is a multipurpose electronic resource for not only participants, but all those interested in the subject. This site contains almost all of the materials used during the meeting including background material, slides, ex-ercises, and a list of principal open problems. A list of projects appropriate for undergraduate students is under development.

Despite the workshop’s full schedule, participants were able to enjoy an afternoon off to explore San Antonio’s celebrated Riverwalk and the Alamo. A group of participants even braved the 60 mile trek to Luckenbach, Texas, but (alas) did not catch a glimpse of Willie Nelson.

References

[1] D. D. Anderson (ed.), Factorization in Integral Domains, Lect. Notes Pure. Appl. Math. 189(1997), Marcel Dekker, New York.

[2] S. T. Chapman, On the Davenport constant, the cross num-ber, and their application to factorization theory, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 171(1995), 167-190.

[3] S. T. Chapman (ed.), Arithmetical Properties of Com-mutative Rings and Monoids, Lect. Notes Pure. Appl. Math. 241(2005), Chapman & Hall/CRC, Boca Raton.

[4] A. Geroldinger and F. Halter-Koch, Non-Unique Fac-torizations: Algebraic, Combinatorial and Analytic Theory, Chapman & Hall, Boca Raton, 2006.

Scott Chapman teaches at Trinity University in San Antonio, TX. He can be reached at schapman@trinity.edu.

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13

21

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32

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2

31

33

21

3

42

4

3

( ) ( ) ( )

− + − +

− +

− +

−

l

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− + − +

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52

21

2

31

m m

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December 2007

By Peter Sarnak

Paul Joseph Cohen, one of the stars of 20th century mathematics, passed away in March, 2007 at the age of 72. Blessed with a unique mathematical gift for solving difficult and central problems, he made fundamental breakthroughs in a number of fields, the most spectacu-lar being his resolution of Hilbert’s first problem — the continuum hypothesis.

Like many of the mathematical giants of the past, Paul did not restrict his atten-tion to any one specialty. To him math-ematics was a unified subject which one could master broadly. He had a deep un-derstanding of most areas, and he taught advanced courses in logic, analysis, dif-ferential equations, algebra, topology, Lie theory, and number theory on a reg-ular basis. He felt that good mathemat-ics should be easy to understand and that it is always based on simple ideas once you got to the bottom of the issue. This attitude extended to a strong belief that the well recognized unsolved problems in mathematics are, firstly, the heart of the subject and, secondly, that they have clear and transparent solutions once the right new ideas and viewpoints are found. This belief gave him courage to work on notoriously difficult problems throughout his career.

Paul’s mathematical life began early. As a child and a teenager in New York City he was recognized as a mathemati-cal prodigy. He excelled in mathemat-ics competitions, impressing everyone around him with his rare talent.

After finishing high school at a young age and spending two years at Brooklyn College he went to graduate school in mathematics at the University of Chica-go. He arrived there with a keen interest in number theory, which he had learned by reading some classic texts. There he got his first exposure to modern math-ematics and it molded him as a math-ematician. He tried working with André Weil in number theory but that didn’t pan out. Instead, he studied with Antoni Zygmund, writing a thesis in Fourier Series on the topic of sets of uniqueness.

In Chicago he formed many long last-ing friendships with some of his fellow students (John Thompson, for example, who remained a lifelong close friend).

The period after he graduated with a PhD was very productive: he enjoyed a series of successes in his research. He solved a problem of Walter Rudin’s in Group Algebras and soon after that he obtained his first breakthrough on what was considered to be a very difficult problem — the Littlewood Conjecture. He gave the first nontrivial lower bound for the L1 norm of trigonometric poly-nomials on the circle whose Fourier co-efficients are either 0 or 1. The British number theorist-analyst Harold Daven-port wrote to Paul saying that if Paul’s proof held up, he would have bettered a generation of British analysts who had worked hard on this problem. Paul’s proof did hold up; in fact, Davenport was the first to improve on Paul’s result. This was followed by work of a num-ber of people, with the complete solu-tion of the Littlewood Conjecture being achieved separately by Konyagin and Mcgehee-Pigno-Smith in 1981.

In the same paper, Paul also resolved completely the idempotent problem for measures on a locally compact abelian group. Both of the topics from this paper continue to be very actively researched today especially in connection with ad-ditive combinatorics.

As an instuctor at MIT, Paul was in-troduced to the question of uniqueness for the Cauchy problem in linear partial differential equations. Alberto Calderon and others had obtained uniqueness re-sults under some hypotheses, but it was unclear whether the various assump-tions were essential. Paul clarified this much studied problem by constructing examples where uniqueness failed in the context of smooth functions, showing in particular that the various assumptions that were being made were in fact nec-essary. He never published this work (other than putting out an ONR Techni-cal report) but Lars Hörmander incor-porated it into his 1963 book on linear partial differential equations, so that it became well known. This was one of many instances in which Paul’s impact on mathematics went far beyond his published papers. He continued to have a keen interest in linear PDEs and taught graduate courses and seminars on Fou-rier Integral Operators during the 70s, 80s, and 90s.

After spending some time in Princeton at the Institute for Advanced Study, Paul moved to Stanford in 1960; there he re-mained for the rest of his life. He said that getting away from the lively but hectic mathematical atmosphere on the East Coast allowed him to sit back and think freely about other fundamental problems. He has described in a num-ber of places (see, for example [Yan-dell]) his turning to work on problems in the foundations of mathematics. By 1963 he had produced his proof of the independence of the continuum hypoth-esis, as well as of the axiom of choice, from the axioms of set theory. His ba-sic technique to do so, that of “forc-ing,” revolutionized set theory as well as related areas. To quote Hugh Woodin in a recent lecture “it will remain with us for as long as humans continue to think about mathematics and truth.” A few years later Paul combined his inter-est in logic and number theory giving a new decision procedure for polynomial equations over the p-adics and the reals. His very direct and effective solution of

Remembering Paul Cohen

Paul Cohen. Photograph courtesy of Stanford News Service.

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MAA FOCUS December 2007

the decision problem has proved to be central in many recent developments in the subject [Macintyre].

After 1970 Paul published little, but he continued to tackle the hardest prob-lems, to learn and to teach mathemat-ics and to inspire many generations of mathematicians. I was a member of the new generation who was lucky enough to be around Paul.

I first heard of Paul when I was still an undergraduate in Johannesburg in the 1970s. I was taken by the works of Gauss, Dirichlet and Riemann, but studying with them was not an option. However Paul, whose work on the con-tinuum problem is recorded in any good introductory text to mathematics, was apparently alive and well and living in California. Moreover his reputation and stories of his genius had reached all corners of the mathematical world. Kathy Driver (then Kathy Owen and to-day Dean of Science at the University of Cape Town) had just returned from Stanford with the following sort of de-scription of Paul, which she e mailed me recently.

“Paul was an astonishing man. Impa-tient, restless, competitive, provocative and brilliant. He was a regular at cof-fee hour for the graduate students and the faculty. He loved the cut-and -thrust of debate and argument on any topic and was relentless if he found a logical weakness in an opposing point of view. There was simply nowhere to hide! He stood out with his razor-sharp intellect, his fascination for the big questions, his strange interest in “perfect pitch” (he brought a tuning fork to coffee hour and tested everyone) and his mild irritation with the few who do have perfect pitch. He was a remarkable man, a dear friend who had a big impact on my life, a light with the full spectrum of colours.”

I set my goal to study with Paul in the foundations of mathematics and was lucky enough to get this opportunity. Paul lived up to all that I expected. Soon after I met him he told me that his inter-ests had moved to number theory and in particular the Riemann Hypothesis, and

so in an instant my interests changed and moved in that direction too.

Given his stature and challenging style he was naturally intimidating to students (and faculty!). This bothered some; it is probably the reason he had few graduate students over the years. I have always felt that this was a pity, because one could learn so much from him (as I did) and he was eager to pass on the wealth of understanding that he had acquired. Once one got talking to him he was al-ways very open and welcomed with en-thusiasm and appreciation your insights when they were keener than his (which I must confess wasn’t that frequent). As a student I could learn from him results in any mathematical area. Even if he didn’t know a particular result he would eagerly go read the original paper (or, I should say, he would skim the paper, of-ten creating his own improvised proofs of key lemmas and theorems) and then rush back to explain it.

His ideas on the Riemann Hypothesis (about which I will write elsewhere) led him to study much of the work of

Atle Selberg and especially the “trace formula”. This became my thesis topic. Paul and I spent one or two years go-ing through this paper of Selberg and providing detailed proofs of the many theorems that were announced there. We wrote these up as lecture notes, which both Paul and I used repeatedly over the years as lecture notes for classes that we taught. Sections of these notes have found their way into print (in some cases with incorrect attribution) but un-fortunately we never polished them for publication.

Paul continued to work on the Riemann Hypothesis till the end, not for the glory, but because he believed in the beauty of the problem and expected that a solution would bring a deep new understanding of the integers. As mentioned above, it was his strong belief that such problems have simple solutions once properly understood. This gave him the courage to continue this life long pursuit. When working on such problems one is out there alone, with nothing to fall back on. Most professional mathematicians sim-ply don’t take this kind of risk.

At Stanford Paul and his wife Christi-na hosted many dinners and parties for students (graduate and undergraduate), faculty and visitors to the department. I remember many occasions where Paul would treat a visitor to a personal guid-ed tour of San Francisco and the bay area. This opening of their house and their hospitality is remembered fondly by many mathematicians around the world.

Paul’s passing marks the end of an era at Stanford. The world has lost one of its finest mathematicians and for the many of us who learned so much from him and spent quality time with him, it is dif-ficult to come to terms with this loss.

References

B. H. Yandell, The Honors Class, A.K. Peters, Ltd. 2002, pages 59--84.

A. Macintyre, in records of proceedings at meetings of the London Mathematical Society meeting, June 22, 2007.

The Stuyvesant Math Team, Fall 1948. Top row, left to right: Unknown, Un-known, Elias Stein, Harold Widom, Paul Cohen. Bottom row: Unknown, Martin Brilliant, Unknown. Photograph cour-tesy of Martin Brilliant, from Indicator, January 1949.

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December 2007

A few years ago, Ben Fusaro asked me what I thought was the most impor-tant college course for an undergraduate. I replied in an instant: Statistics. Ben agreed. His innovations in mathematical modeling and environmental mathemat-ics appeal to our advanced students. Sta-tistics, on the other hand, could help all undergraduates in both their future per-sonal and professional lives.

In 2005, the Department of Education released a report that revealed mathe-matics to be the most serious obstacle to college graduation in the United States. The report described mathematics as an “insurmountable barrier” for economi-cally disadvantaged college students. We should be looking to fix this.

We can make a difference in student retention, effect long-term influence to-ward our students, and share an enjoy-able, almost fun, experience. Students in my statistics courses are successful. Concordia College–NY has wonder-ful students with mean SAT scores in mathematics at approximately the na-tional average, but this approach could be replicated at any college. I came up with many of these strategies during my eighteen years teaching statistics at SUNY–Westchester Community Col-lege. The resulting course was enjoy-able for the students and for me.

Here is how. I rely on four moves that are the results of two decades of experi-ence.

Shorten the List of Topics

If I did not use a topic from statistics in my first five published interdisciplin-ary books or my consulting work (both college and legal), then I did not teach that topic. We are still left with seven or eight key topics: probability, small and large sample confidence intervals, sam-ple size, one and two sample hypothesis testing of the difference of means and proportions, correlation, and linear re-gression. That is plenty. Postpone anova,

multiple regression, and nonparametric statistics for a second, optional course.

Some might feel that this amounts to lowering standards. But consider the collateral damage of finishing the book. Do you really reach the students by cov-ering a different topic every day with little time to apply the concept to the student’s future personal and profes-sional life? Are we denying students an opportunity to become an English teach-er or social worker because he/she can-not understand Chebyshev’s Theorem?

I confess to ignorance as to where Iraq is on the modern globe. My high school and college French were not adequate for me to obtain routine directions while in Paris. We believe that students need arithmetic and algebra as life skills. As a consequence, we frequently end up de-nying them a college degree, which is essential for economic survival. Are we abusing our power by maintaining the status quo without scrutinizing the role mathematics plays in winnowing the preferred jobs?

Rely on the TI-83 Calculator

Many (if not most) of our students can-not read their textbook. Some students will never learn elementary algebra, despite our best efforts. The calcula-tor compensates for their mathematical handicaps. I wrote a 92 page book, Es-sentials of Elementary Statistics (TI-83 Based), that I give each student for free. For every topic I teach in statistics, my book shows the students the step-by-step key strokes to perform the statistical analysis. This becomes my main text for statistics. Write me and I will send you a copy of my book, that you can duplicate and distribute to your students, until that blessed day when (or if) it is published. I assign Triola’s masterful Elementary Statistics for additional homework and as an invaluable lifelong reference text.

Have Fun

I assign three interdisciplinary books

and require students to write essays and attend book discussions on the readings. If they fail to attend the discussion (with the essay as ticket for admission), they lose a letter grade. My assigned books are Man’s Search for Meaning (Frankl), Our Neglect, Denial, and Fear (Loase), and The Millionaire Next Door (Stanley and Danko). The three essays, which go through a revise-rewrite cycle, coupled with a required short research paper, constitute 1/3 of the students’ grade. Each student is required to write a two page (400-500) word essay critiquing each book and discussing the statistical issues that I raise from each book.

Man’s Search for Meaning is one of the twentieth century’s great works. I relate statistics to the book by having each student complete a survey to quantify students’ perceptions of their positive influence toward others. We discuss this concept at the book seminars. As a mathematician-psychologist, the books I assign may be different from the ones you assign. Statistics has the potential to prepare students for their future person-al and professional life with relevance that few courses can rival, and using it to think about books such as these high-lights that potential.

If you find the readings interesting, your students may as well. At worst, they will be amazed by our idiosyncratic tastes. Experiment. Have fun. If you are not having fun, your students are not.

I highly recommend The Millionaire Next Door, an outgrowth of extensive data mining and statistical analysis, as one of your required readings. It fo-cuses on money and how to become rich in America — a high interest topic for everyone. I suggest to the students that they advise their “Baby Boomer” parents about retirement, using material from the book. I also caution them that their parents do not expect us to teach their children anything that could help them. Be gentle with parents with weak hearts: the shock of being counseled by their children may be too much to bear.

Statistics: A Key to Student Success in College and LifeBy John Loase

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MAA FOCUS December 2007

Allow Students In

If a student places in college arithmetic, which is in fact a euphemism for fourth grade arithmetic, the conditional prob-ability of that student graduating from college within five years is in the same ballpark as one of us becoming a major league athlete. We are no better than the elementary, junior high, and senior high mathematics teachers, who failed to im-part the fundamentals of arithmetic, and they had 180 days a year. We have 45–60 class sessions. Yes, some students may earn a C in statistics without having mastered addition of fractions. But that student now has an improved chance of graduating from college, earning a de-cent living, and hiring an accountant to do his/her taxes.

I consulted at one college that had a failure rate far in excess of 50% among

its educationally disadvantaged stu-dents. At another college, the Director of Counseling regularly informed me that the high rate of remedial mathemat-ics failure was cutting the graduation rate in half. Effective remedial courses are costly, resource intensive, and only available to a small fraction of the un-der-prepared mathematics students. Those same students could succeed in a statistics course.

Few professions scrutinize the fairness of their credentialing system, which cre-ates winners and losers. In fact, several professions increase their mathematics requirements whenever supply exceeds demand. Several years ago, I attended an MAA session on “Why is Math so Unpopular?” After an hour of missing the painful truth, I raised my hand and asked whether math was so unpopular because we mathematicians serve as the

students’ executioners, by whom they are screened out of Engineering, Medi-cine, and MBAs. Mathematicians have the tools and the intellect to scrutinize our largely unchallenged assumptions about what constitutes essential math-ematical knowledge for college gradu-ates. We could lead other professions in a spirit of societal transformation.

We can prepare all our students for their future professional challenges by tap-ping the vitality of statistics. We need to prune the curriculum, add enrichment activities, empower students with the TI-83 Calculator, and (if remediation does not work) give students a chance to succeed in statistics.

John Loase is Chair of the Mathematics Department at Concordia College-NY.

It is Time to Support ExtremesBy Reza Noubary

Traditionally, statistics has focused on the study of values with high frequen-cies and on averages. This becomes clear when we examine textbooks and course descriptions for introductory statistics courses. In today’s world, however, it is not sufficient to focus on averages alone. It has become important and even necessary to study extremes and rare events, since they are usually critical, newsworthy, and are often accompanied by severe consequences.

The celebrated central limit theorem has given statistics its focus on averages — we do what we know how to do. The statistical theories of extremes are less simple, less unified, and more recent. However they are not less important and there is a need for their inclusion in in-troductory statistics courses.

Why Averages?

Most of the elementary mathematics we teach deals with the “smooth” part of the world. In the same way most classical statistical methods are based on the idea of smoothing the data as a

part of their analysis. Specifically, some methods treat the data as a message and seek to decompose it into a systematic (deterministic, trend, signal) part and a random (stochastic, noise) part.

When the form of the systematic part is known, it is easy to separate it from the random part. In the absence of in-formation pertaining to either system-atic or random parts, smoothing is used for separation. This is usually carried out assuming that the systematic part is smooth and the random part is rough. Smoothing is an exploratory operation, a means of gaining insight into the na-ture of data without precisely-formu-lated models or hypotheses. Smooth-ing is often achieved by some sort of averaging (low-pass filtering). Once the smooth part is determined, the dif-ference between the original message and the smooth part is used to present the rough part. The rough part is usu-ally utilized to make reliability state-ments regarding the systematic part. For data with a time index (time series), one popular smoothing technique is the so-called moving average. The idea is to

average the neighboring values and to move it along the time axes.

Why Extreme Values?

In many applications, it is not appropri-ate to focus on averages alone. In fact, there are many instances where extreme values and values with low frequen-cies are of more concern. Examples in-clude: large natural disasters compared to moderate ones; weak components or links compared to their average counter-parts; large insurance claims compared to average claims.

Extreme values are usually analyzed using one of the three major theories: The “Extreme Value Theory” that deals with maxima or minima of the subsam-ples, the “Threshold Theory” that deals with values above or below a specified threshold, and the “Theory of Records” that deals with values larger (smaller) than all the previous values.

These theories deal with the actual values of the extremes. The frequen-cies of extremes are analyzed using the

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“Theory of Exceedances.” This theory deals exclusively with number of times a chosen threshold is exceeded. In most applications, the number of exceedances and values of excesses over a threshold are combined to yield a more detailed analysis. For example, insurance com-panies analyze both the number of times a “large” claim is made and the amount by which these claims exceed a specific large threshold. Reliability engineers study both the number of extreme loads, such as large earthquakes, and their magnitudes.

Extreme Value Theory

Extreme value theory deals with annual (or any other periodic) maxima or min-ima. Specifically, the theory is based on dividing the sample into a subsamples and fitting a distribution for maxima or minima of the subsamples. For example, data may consist of largest earthquakes in California for each of the last 100 years.

As in most statistical theories, first the distribution of the largest or the smallest values was derived for a finite sample. Then, by letting sample size tend to infinity, the limiting distribution of ex-treme values was obtained. For this a typical maximum Yn is reduced with a location parameter β

n and a scale pa-

rameter αn (assumed to be positive)

such that the distribution of standard-ized extremes (Yn – β

n) /α

n is non-de-

generate. The forms of the limiting dis-tributions are specified by the extreme value theorem. This theorem states that there are three possible types of limit-ing distributions for maxima. They are known as the Gumbel distribution (Type I), the Fréchet distribution (Type II), and the Weibull distribution (Type III). These three forms can be combined to yield the so-called generalized extreme value distribution.

Most classical distributions fall in the domain of attraction of one of these three types. For example, the distribu-tion of maxima of samples from a nor-mal distribution tends to the Gumbel distribution. More generally, the neces-sary and sufficient conditions for a par-ticular distribution to fall in domain of

attraction of one of the three types are known. Only distributions unbounded to the right can have a Fréchet distribu-tion as a limit. Only distributions with finite right end point can have a Weibull as a limit. The Gumbel distribution can be the limit of bounded or unbounded distributions.

How do we decide which of the three limiting distributions to fit to the data? Theoretically, we can use the fact that each of the classical distributions falls in the “domain of attraction” of one of the limiting distributions above. This works if distribution of the original data is known. Unfortunately, in practice such information is not usually available and decisions should be based on the area of application or on expert opinion. When information about the appropriate lim-iting distribution is absent, statistical goodness-of-fit may be used.

Since extreme value distributions are fitted to, for example, annual maxima or minima, some relevant data related to the years with several large observed values could be discarded and some less informative data related to the years with no real large values could be re-tained. The threshold theory discussed next avoids this problem.

Threshold Theory

Threshold theory allows one to make inferences about the values above or be-low a threshold, that is, the upper or the lower tails of a distribution. It considers the excesses, the differences between the observations over the threshold and the threshold itself. Like extreme value distributions, there are three models for tails. They are long tail Pareto, medium tail exponential, and short tail distribu-tion with an end point.

Again, most classical distributions fall in domain of attraction of one these tail models. It has been shown that the natu-ral parametric family of distributions to consider for excesses (tail) is the Gen-eralized Pareto Distribution (GPD). In practice the proposed method is to treat the excesses as independent random variables and to fit the GPD to them. The choice of threshold is, to a large ex-

tent, a matter of judgment depending on what is considered large or small.

The theory is very useful when model-ing large values based on observed large values is of main concern. Clearly the modeling and prediction of large earth-quakes should be based on past large earthquakes not on the medium or small earthquakes.

Theory of Records

The theory of records deals with values that are strictly greater than or less than all previous values. Usually the first val-ue is counted a record. Then a value is a record (upper record or record high) if it is bigger than all previous values. The study of record values, their frequen-cies, times of their occurrences, their distances from each other, etc. consti-tutes the theory of records.

Formally the theory deals with four main random variables: the number of records in a sequence of n observations, the re-cord times, the waiting time between the records, and the record values. It is interesting to note that the first three can be investigated using non-parametric or distribution-free methods whereas the last one requires parametric methods.

Records in general and sports records in particular are of great interest, and their occurrence usually results in a great deal of excitement and the media attention.

Traditionally, data values with high fre-quencies and averages have been the focus of statistical analysis and model-ing. This has rested on the celebrated central limit theorem. Extremes, rare events, and records values have been mostly avoided and often treated as out-liers rather than important information. But low-frequency high-consequence events are very important. We think that there is a need to introduce students to these vital concepts.

Reza Noubary teaches at Bloomsburg University, in Bloomsburg, PA. His re-search interests include time series analysis, geostatistics, reliability, risk analysis, and applications of mathemat-ics in sports.

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EMPLOYMENT OPPORTUNITIES

ALASKA

University of Alaska Southeast (Juneau)Assistant or Associate Professor of Mathematics at the University of Alaska Southeast (Juneau): This is a bipartite (teaching and service), tenure-track position beginning August 2008. The position will remain open until filled; however, first consideration will be giv-en to applications received by January 11, 2008. Members of the search com-mittee will participate in the Employ-ment Center at the 2008 Joint Meeting. Go to www.uakjobs.com/applicants/Central?quickFind=60450 for position announcement and for information about the UAS Mathematics Program go to http://www.uas.alaska.edu/math/. UAS is an AA/EO Employer and Edu-cational Institution.

ARKANSAS

Henderson State UniversityMathematics, Assistant Professor or Instructor: Doctorate in mathematics, mathematics education or related field preferred; master’s degree in same re-quired; teach full range of undergraduate courses. For more information and ap-plication procedures, please visit www.hsu.edu/Affirmative-Action. Henderson State University, Arkadelphia AR is an affirmative action/ADA, EOE. Women and minorities are especially encour-aged to apply.

CANADA

McGill UniversityThe Department of Mathematics and Statistics invites applications for a ten-ure-track position in discrete mathemat-ics or continuous optimization. While the appointment is expected to be made at the level of an Assistant Professor, the Department would consider applicants for a senior position. The candidate must have a doctoral degree at the date of appointment. They are also expected to have demonstrated the capacity for independent research of excellent qual-ity. Selection criteria include research accomplishments, as well as potential

contributions to the educational pro-grams of the Department at the graduate and undergraduate levels.

Applications with a curriculum vitae, a list of publications, a research outline, an account of teaching experience, a state-ment on teaching, and the names, phone numbers and e-mail addresses of at least four references (with one addressing the teaching record) should be sent to:

Professor Bruce Shepherd Chair, Discrete Mathematics or Contin-uous Optimization Search CommitteeDepartment of Mathematics and StatisticsMcGill University805 Sherbrooke Street West Montreal QC H3A 2K6 Canada

Candidates must arrange to have the let-ters of recommendation sent directly to the above address. Candidates are en-couraged to include copies of up to three selected reprints or preprints with their applications. To ensure full consider-ation, applications must be received by January 15, 2008.

To facilitate notification of the outcome of the search, candidates should send an email to the address DMCOSearch08@math.mcgill.ca at the time of applica-tion.

McGill University is committed to eq-uity in employment and diversity. It welcomes applications from indige-nous peoples, visible minorities, ethnic minorities, persons with disabilities, women, persons of minority sexual orientations and gender identities and others who may contribute to further diversification. All qualified applicants are encouraged to apply; however, in accordance with Canadian immigration requirements, priority will be given to Canadian citizens and permanent resi-dents of Canada.

CONNECTICUT

FairfieldUniversityThe Department of Mathematics and Computer Science at Fairfield Univer-

sity invites applications for three tenure track assistant professorships, to begin in September 2008. A doctorate in math-ematics is required. A solid commitment to teaching, and strong evidence of re-search potential, are essential. We are looking for (1) one person who will be expected to conduct research with un-dergraduate students and (2) two people who will be expected to teach some courses in our graduate program. Grad-uate courses include, but are not limited to, year-long sequences in Abstract and Linear Algebra, Applied Mathemat-ics, Financial Mathematics, Real and Complex Analysis, and Probability and Statistics. In addition, the successful candidates will share a willingness to participate in the university’s core cur-riculum, which includes two semesters of mathematics for all undergraduates.

Fairfield University, the Jesuit Univer-sity of Southern New England, is a com-prehensive university with about 3,200 undergraduates and a strong emphasis on liberal arts education. The department offers a BS and an MS in mathematics. The MS program is an evening program and attracts students from various walks of life – secondary school teachers, eventual Ph.D. candidates, and people working in industry, among others. The teaching load is 3 courses/9 credit hours per semester and consists predominantly of courses at the undergraduate level.

Fairfield offers competitive salaries and compensation benefits. The picturesque campus is located on Long Island Sound in southwestern Connecticut, about 50 miles from New York City. Fairfield is an Affirmative Action/Equal Opportu-nity Employer. For further details see http://cs.fairfield.edu/mathhire. Appli-cants should send a letter of applica-tion, a curriculum vitae, teaching and research statements, and three letters of recommendation commenting on the applicant’s experience and promise as a teacher and scholar, to Matt Coleman, Chair of the Department of Mathemat-ics and Computer Science, Fairfield University, Fairfield CT 06824-5195. Please indicate in your cover letter the position for which you are applying.

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Full consideration will be given to com-plete applications received by January 15, 2008. We will be interviewing at the Joint Mathematics Meetings in San Di-ego, January 6-9. Please let us know if you will be attending.

DISTRICT OF COLUMBIA

American UniversityTenure-track Assistant Professor in the Mathematics and Statistics Department at American University, beginning Fall 2008. Qualified candidates will have a strong background in Mathematics or Statistics. Ph.D. and teaching expe-rience required. American University is an EEO/AA employer. Minority and women candidates are encouraged to apply. See math.american.edu/positions, or contact the Department of Mathemat-ics and Statistics at (202) 885-3120 for details.

FLORIDA

Lakeland CollegeAssistant Professor of Mathematics Lakeland College, an independent lib-eral arts institution located in northeast-ern Wisconsin, invites applications for a full-time, tenure-track position of Assis-tant Professor – Mathematics, beginning in August, 2008. The candidates should have the ability to teach a wide range of undergraduate courses in Mathematics and the ability to teach some computer science courses will be an advantage. The candidates should be committed to quality teaching and advising and be willing to participate in the College’s interdisciplinary general education pro-gram.

A PhD in mathematics is required. Some college teaching experience is preferred. Compensation will be commensurate with background and experience and a comprehensive benefit program is of-fered.

Interested candidates should submit a letter of interest, current resume, a state-ment of teaching philosophy and three current letters of recommendation to: Director of Human Resources, Lakeland College, P.O. Box 359, Sheboygan, WI. 53082-0359; or email to: schoemerjr@

lakeland.edu. Review of applications will begin January 28, 2005 and will continue until the position is filled. For additional information on Lakeland Col-lege, please access our web site at www.Lakeland.edu.

An affirmative action / equal opportu-nity employer

IOWA

University of IowaActuarial science tenure-track assistant professor starting 8/08. Ph.D. required 8/20/08. Actuarial Fellowship or Asso-ciateship highly preferred. Industrial ex-perience helpful. Duties include teaching and research in actuarial science and/or financial mathematics, involvement in Ph.D. program, and supervision of Ph.D. students. Selection begins 12/03/07. CV, three reference letters, and transcript for recent Ph.D.s to Actuarial Search, Sta-tistics & Actuarial Science, Univ. of Iowa, Iowa City, IA 52242-1409.actu-arial-search@stat.uiowa.edu

Women, minorities encouraged to apply. AA/EOE.

MAINE

Colby CollegeMathematics DepartmentThe Department of Mathematics at Col-by College invites applications for a one-year sabbatical replacement position in mathematics at the assistant professor or instructor level, beginning September 1, 2008. Ph.D. in mathematics preferred; A.B.D. considered. Five course teaching load. Evidence of exceptional teaching ability is required. The ability to teach a course in the history of mathematics is desirable but not required.

Send curriculum vitae, a statement on teaching and research, and three letters of recommendation (all in hard copy) to: Mathematics Search Chair, Depart-ment of Mathematics, Colby College, 5830 Mayflower Hill, Waterville, ME 04901. We cannot accept applications in electronic form. Review of applications will begin on January 15, 2008 and will continue until the position is filled.

Colby is a highly selective liberal arts college located in central Maine. The college is a three-hour drive north of Boston and has easy access to lakes, skiing, the ocean, and other recreational and cultural activities. For more infor-mation about the position and the de-partment, visit our web site at www.colby.edu/math.

Colby is an Equal Opportunity/Affirma-tive Action employer, committed to ex-cellence through diversity, and strongly encourages applications and nomina-tions of persons of color, women, and members of other under-represented groups. For more information about the College, please visit the Colby Web site at www.colby.edu.

MARYLAND

Montgomery CollegeA Learning CollegeOffice of Human ReaourcesYour Resource to ExcellenceConsider Montgomery College for your next career move!

Montgomery College, one of Mary-land’s oldest community colleges, is a multi-campus institution located in Montgomery County, Maryland. It has earned a reputation as one of the leading community colleges in the nation by ap-plying a winning formula: excellence in teaching; innovative partnerships; and unrivaled leadership.

Montgomery College is seeking a full-time Mathematics faculty member for the Rockville Campus. Assignment be-gins August, 2008. For more details and to apply online, please visit: https://jobs.montgomerycollege.edu. Starting salary range is $44,096 to $63,596 per year. Online applications must be received by 12 noon, Tuesday January 2, 2007.

Mathematics Faculty – Position #2688 – Rockville Campus

Required Qualifications: • A Master’s Degree or Ph.D. in Math-ematics, Mathematics Education, or a

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MAA FOCUS December 2007

strongly related field. • At least 18 semester hours of gradu-ate level courses in mathematics. • Coursework that supports the can-didate’s capacity to teach the spectrum of courses offered by the Mathematics Department. • At least one year of recent full-time experience (or the equivalent) teaching mathematics courses typical of commu-nity college offering. Desirable Qualifications : • Involvement in programs for improv-ing student success. • Experience in using graphing calcu-lators. • Experience using computer algebra systems such as Matlab, Maple or Math-ematica in mathematics courses.Application Procedures: In order to be considered for the posi-tion, you must complete the following steps: • Create the Montgomery College on-line employment application. To create an application and to apply on-line, please visit: https://jobs.montgom-erycollege.edu• After you complete your application, please be sure to search jobs and click on the Rockville Campus Mathematics position and apply to the job.

PLEASE NOTE: • The online application deadline is 12 noon, Wednesday, January 2, 2008. • Incomplete applications will not be considered.Office of Human Resources – 240-567-5353. Montgomery College is an equal oppor-tunity employer committed to fostering a diverse academic community among its student body, faculty, and staff.

MASSACHUSETTS

Williams CollegeWilliams College Department of Math-ematics and Statistics invites applica-tions for a newly authorized visiting po-sition in mathematics for the 2008-2009 year, at the rank of assistant professor. A Ph.D. is required. Send a vita and three letters of recommendation on teaching and research to: Visitor Hiring Com-mittee, Department of Mathematics and Statistics, Williams College, William-

stown, MA 01267. Consideration of applications will begin on November 15th and continue until the position is filled. Williams College is dedicated to providing a welcoming intellectual en-vironment for all of its faculty, staff and students; as an AA/EOE employer, Wil-liams especially welcomes applications from women and minority candidates.

Williams CollegeThe Williams College Department of Mathematics and Statistics invites ap-plications for one tenure track position in mathematics, beginning fall 2008, at the rank of assistant professor (in an exceptional case, a more advanced ap-pointment may be considered). We are seeking a highly qualified candidate who has demonstrated excellence in teaching and research, and who will have a Ph.D. by the time of appointment.

Williams College is a private, residen-tial, highly selective liberal arts college with an undergraduate enrollment of ap-proximately 2,000 students. The teach-ing load is two courses per 12-week se-mester and a winter term course every other January. In addition to excellence in teaching, an active and successful re-search program is expected.

To apply, please send a vita and have three letters of recommendation on teaching and research sent to the Hiring Committee, Department of Mathemat-ics and Statistics, Williams College, Williamstown, MA 01267. Teach-ing and research statements are also welcome. Evaluation of applications will begin on or after November 15 and will continue until the position is filled. Williams College is dedicated to providing a welcoming intellectual environment for all of its faculty, staff and students; as an EEO/AA employer, Williams especially encourages applica-tions from women and minorities. For more information on the Department of Mathematics and Statistics, visit http://www.williams.edu/Mathematics.

NEW HAMPSHIRE

Dartmouth CollegeJohn Wesley Young Research Instruc-torship

The John Wesley Young Instructorship is a postdoctoral, two- to three-year ap-pointment intended for promising Ph.D. graduates with strong interests in both research and teaching and whose re-search interests overlap a department member’s. Current research areas in-clude applied mathematics, combinator-ics, geometry, logic, non-commutative geometry, number theory, operator al-gebras, probability, set theory and to-pology. Instructors teach four ten-week courses distributed over three terms, though one of these terms in residence may be free of teaching. The assign-ments normally include introductory, advanced undergraduate, and graduate courses. Instructors usually teach at least one course in their own specialty. This appointment is for 26 months with a monthly salary of $4667, and a possible 12 month renewal. Salary includes two-month research stipend for Instructors in residence during two of the three sum-mer months. To be eligible for a 2008-2011 Instructorship, candidate must be able to complete all requirements for the Ph.D. degree before September, 2008. Applications may be obtained at http://www.math.dartmouth.edu/recruiting/ or http://www.mathjobs.org Position ID: 237-JWY. General inquiries can be di-rected to Annette Luce, Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, New Hamp-shire 03755-3551. At least one referee should comment on applicant’s teaching ability; at least two referees should write about applicant’s research ability. Ap-plications received by January 5, 2008 receive first consideration; applications will be accepted until position is filled. Dartmouth College is committed to di-versity and strongly encourages applica-tions from women and minorities.

NEW YORK

Niagara Universitywww.niagara.edu The Mathematics De-partment at Niagara University, a pri-vate Catholic institution sponsored by the Vincentian Community seeks As-sistant Professor, tenure track, August 2008 start.

Requirements: Strong commitment to undergraduate teaching, ability to do

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December 2007

scholarly research, Ph.D. in Mathemat-ics, Statistics or related field.

Candidate to teach a broad range of classes from introductory statistics to senior seminar; should possess the abil-ity to work with students both inside and outside the classroom, including student research; should be interested in teach-ing mathematics to pre-service teach-ers.

Located near scenic Niagara Falls, Ni-agara University is a predominantly un-dergraduate liberal arts university.

Application letter, vitae and three rec-ommendation letters: Dr. Richard Cra-mer-Benjamin, Chairperson, Mathemat-ics Department, Niagara University, NY 14109-2044. Applications reviewed un-til position filled. AA/EOE

NORTH CAROLINA

Western Carolina UniversityDepartment Head, Mathematics and Computer ScienceWestern Carolina University invites ap-plications for the position of Head of the Mathematics and Computer Science Department. This is a twelve-month position, with a requested start date of July 1, 2008. A doctorate in the Math-ematical Sciences from an appropriately accredited institution, scholarly creden-tials and experience commensurate with appointment at the rank of Full Professor or Associate Professor, and evidence of excellence in teaching, research, and ac-ademic leadership are required. Screen-ing begins January 10, 2008 and contin-ues until the position is filled. To apply visit: https://jobs.wcu.edu/applicants/Central?quickFind=51042. Western Caro-lina University, a campus of the Univer-sity of North Carolina, is a vibrant and growing university, and is an AA/EOE employer that conducts background checks before employment. Proper documentation of identity and employ-ability are required at the time of em-ployment.

Western Carolina UniversityMathematics EducationTenure-track Assistant Professor in mathematics education beginning Au-

gust 2008 in a department that has graduated the western region outstand-ing mathematics education student of the year for NC for each of the past 17 years. Doctorate in mathematics educa-tion or a related field with the equiva-lent of a master’s in mathematics is re-quired from an appropriately accredited institution. Excellence in teaching, a commitment to ongoing research and service are required. Screening be-gins December 10, 2007 and continues until the position is filled. To apply visit: https://jobs.wcu.edu/applicants/Central?quickFind=51041. Western Car-olina University, a campus of the Uni-versity of North Carolina, is an AA/EOE employer that conducts background checks before employment. Proper documentation of identity and employ-ability are required at the time of em-ployment.

PENNSYLVANIA

California University of PATenure-track position at California Uni-versity of PA in MATHEMATICS: Ef-fective August 2008. A PhD in Mathe-matics is required. Teaching experience at the college/university level, and practical experience, such as consult-ing, is preferred. The successful candi-date will have a strong desire to teach, actively recruit students, and partici-pate in the development of web-based courses and/or programs. Candidates should be experienced in the use of in-novative approaches that are student-centered, inquiry-based, and hands-on oriented. For more information about the position, and instructions about how to apply (deadline is January 11, 2008): http://www.cup.edu/employment/index.jsp . Cal U is M/F/D/V/AA/EEO.

SOUTH CAROLINA

Central Carolina Technical CollegeMathematics Instructor (2 positions): #020901MA & #139217MA. Respon-sible for teaching college credit algebra, statistics, trigonometry, geometry and calculus classes at main and off campus locations. Requires Master’s Degree in mathematics or a Master’s Degree with 18 graduate semester hours in math-ematics. Post-secondary teaching ex-

perience with ability to teach all levels of college math and Master’s Degree in mathematics or Master’s Degree with 18 graduate semester hours in mathematics preferred. Apply online at http://jobs.sc.com or visit www.cctech.edu.

Questions? Contact the Personnel Of-fice by email: personnel@cctech.edu or by mail: Central Carolina Technical College 506 N. Guignard DriveSumter, SC 29150EOE/AA

TEXAS

The University of Texas at TylerThe University of Texas at Tyler invites applications for the position of Chair of the Department of Mathematics to begin fall 2008. The university seeks candidates who will energetically lead the department in continuing to build excellent undergraduate and graduate programs and will mentor faculty in teaching, research, and service.

The successful candidate will have a PhD in mathematics, an outstanding record of teaching and research com-mensurate with a tenured faculty ap-pointment, effective leadership, admin-istrative, and interpersonal skills, and the ability to lead the faculty in obtain-ing external funding.

Located 90 miles east of Dallas in the beautiful piney woods of East Texas, The University of Texas at Tyler has an enrollment of about 6000 students. The Department of Mathematics offers de-grees at the undergraduate and graduate levels. For general information about The University of Texas at Tyler, visit www.uttyler.edu. The Department of Mathematics has a web site at http://math.uttyler.edu.

Please submit (electronically as attach-ments, if possible) a letter of applica-tion, curriculum vitae, unofficial tran-scripts, a brief description of research plans, statement of teaching philosophy, statement of leadership philosophy, and names and email addresses of at least four references to Dr. Don Kille-

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MAA FOCUS December 2007

brew, Chair, Department of Mathemat-ics Chair Search Committee, dkille@mail.uttyl.edu. Paper submissions can be sent to Department of Mathematics, The University of Texas at Tyler, 3900 University Blvd., Tyler, Texas 75799.

Review of applications will begin imme-diately and continue until the position is filled. Applicants must be prepared to furnish the university with proof of eli-gibility to work in the United States. UT Tyler is an Equal Employment Opportu-nity/Affirmative Action Employer.

UTAH

Brigham Young UniversityApplications are invited for visiting professorships at any level (assistant, associate, or full professor) in the De-partment of Mathematics at Brigham Young University. The department has a strong commitment to undergraduate education and has a small, but strong, doctoral program.

Qualifications: Ph.D. in mathematics or related field; demonstrated excellence in teaching; desire and ability to teach undergraduate and graduate level math-ematics, including courses at the fresh-man level; desire and ability to conduct high quality mathematical research.

Responsibilities: Teaching mathematics at the undergraduate and graduate level, including courses at the freshman level; conducting and directing independent research; collaborating with other fac-ulty at the university; supporting uni-versity, college, and department goals and missions.

Review of applications will begin on December 1, 2007 and will continue until the position is filled. To apply, go to http://yjobs.byu.edu. For further in-formation about the department and the university go to http://math.byu.edu.

Brigham Young University is an equal opportunity employer. Preference is given to qualified candidates who are members in good standing of the affiliated church, The Church of Jesus Christ of Latter-day Saints.

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Follow the journey of the six high school students who represented theUnited States in 2006 at the world’s toughest math competition – theInternational Mathematical Olympiad (IMO).

Hard ProblemsHard Problems gives you behind-the-scenes access to an extraordinary groupof individuals. Enter a world of dedication, perseverance, and rigorouspreparation. Experience the joys and frustrations related to trying to solveincredibly challenging math problems against competitors from 90 countriesand against the clock.

®

The Mathematical Association of AmericaInvites you to

The Exclusive World Premiere Screening of

Hard ProblemsThe Road to the World’s Toughest Math Contest

a film by George Csicsery

Join the MAA for a reception and the world premiere

Tuesday, January 86:00-8:00 p.m.

please refer to the JMM program for room location

Stop by and meet George Csicsery at theAmerican Mathematics Competitions BoothWednesday, January 9 from 9:00-11:00 a.m.

Pick up your copy of the DVD at the MAA Publications Booth for the discounted priceof $17.50* at anytime during the meeting!

All prices final, sale only good during 2008 Joint Mathematics Meetings in San DiegoPrice includes 7.75% sales tax

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MAA FOCUS December 2007

The Mathematical Association of America1529 Eighteenth St., NWWashington, DC 20036

Periodicals Postage paid at Washington, DC and

additional mailing offices

Would they also love a full-tuition scholarship for a master’s degree in mathematics education, a New York State Teaching Certificate, and a $90,000 stipend in addition to a competitive salary as a New York City secondary school math teacher? Math for America offers fellowships with these benefits and more to individuals who know and love math, enjoy working with young people, have excellent communications skills, and a strong interest in teaching. Log on to learn more.

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